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Set Partitioning: A Survey
Abstract
This paper discusses the set partitioning or equality-constrained set covering problem. It is a survey of theoretical results and solution methods for this problem, and while we have tried not to omit anything important, we have no claim to completeness. Critical comments pointing out possible omissions or misstatements will be welcome.Part 1 gives some background material. It starts by discussing the uses of the set partitioning model; then it introduces the concepts to be used throughout the paper, and connects our problem to its close and distant relatives which play or may play a role in dealing with it: set packing and set covering, edge matching and edge covering, node packing and node covering, clique covering. The crucial equivalence between set packing/partitioning and node packing problems is introduced.Part 2 deals with structural properties of the set packing and set partitioning polytopes. We discuss necessary and sufficient conditions for all vertices of the set packing polytope to be intege...
... In the sequel, we encounter both set and integer partitioning problems. Each has been extensively studied [19,97,62,148] and can be viewed as a particular combinatorial optimization problem [124,116]. We mention that neither is exactly the well-known "partition" problem described by Karp in his classic paper [110,71]. ...
... Such a partition is sometimes called unlabeled to distinguish it from an allocation, which has a prespecified number of elements [97]. For the many applications of these problems, see [19] and [97]. Dorfman's procedure partially motivated one historical line of work [90,95,99,97]. ...
... The basic difficulty is that the number of partitions of a finite set of size n, the so-called nth Bell number [22,153], grows quickly with n. Still, these problems have standard integer linear programming formulations when the objective is additive [19,155,156]. Also, several other structured objectives have been studied [95,9,97,121]. ...
We study Dorfman's classical group testing protocol in a novel setting where individual specimen statuses are modeled as exchangeable random variables. We are motivated by infectious disease screening. In that case, specimens which arrive together for testing often originate from the same community and so their statuses may exhibit positive correlation. Dorfman's protocol screens a population of n specimens for a binary trait by partitioning it into nonoverlapping groups, testing these, and only individually retesting the specimens of each positive group. The partition is chosen to minimize the expected number of tests under a probabilistic model of specimen statuses. We relax the typical assumption that these are independent and indentically distributed and instead model them as exchangeable random variables. In this case, their joint distribution is symmetric in the sense that it is invariant under permutations. We give a characterization of such distributions in terms of a function q where q(h) is the marginal probability that any group of size h tests negative. We use this interpretable representation to show that the set partitioning problem arising in Dorfman's protocol can be reduced to an integer partitioning problem and efficiently solved. We apply these tools to an empirical dataset from the COVID-19 pandemic. The methodology helps explain the unexpectedly high empirical efficiency reported by the original investigators.
... |{n : z tn = k}| ≤ 1 ∀k > 0, ∀t f t = |{n : z tn = 0}| ∀t a t = |{k > 0 : z tn = k and z t n = k for all t < t}| ∀t λ t = |{k > 0 : z tn = k and z t n = k for all t > t}| ∀t a t + d t = |{n : z tn > 0}| ∀t (3) Association hypotheses that do not satisfy these constraints have zero probability. We note that the space of possible associations is exponential in time T and factorial in the number of observations N t at each time t [26]. ...
... Input : x, z, M, y Output : x , z , M 1 Let x = x, z = z 2 Let k = 1 + K(z) 3 Define gather times τ 0 = {t : z tn = 0 for any 1 ≤ t ≤ T } 4 for t = min τ 0 , . . . , max τ 0 do 5 if rand(0, 1) < δ continue 6 Sample p(z tn = k) ∝ p(y tn |x t k , z tn = k) I(z tn = 0) Following Algorithm 3, the Disperse proposal simply chooses an existing object at random (Line 2), removes all its associations by setting them to clutter (Line 3) and deletes the trajectory values for that object (Line 4). ...
... Multi-object tracking can be formulated in several common ways: as a set partitioning problem [3], a set packing problem [20,40], a maximum-weight independent set problem [25] or a multidimensional assignment problem [28]. A review of these formulations is conducted by [11], who shows that the multidimensional assignment formulation is not limited to pairwise terms as the set-packing, network-flow solutions [27,5] are. ...
Robust data association is critical for analysis of long-term motion trajectories in complex scenes. In its absence, trajectory precision suffers due to periods of kinematic ambiguity degrading the quality of follow-on analysis. Common optimization-based approaches often neglect uncertainty quantification arising from these events. Consequently, we propose the Joint Posterior Tracker (JPT), a Bayesian multi-object tracking algorithm that robustly reasons over the posterior of associations and trajectories. Novel, permutation-based proposals are crafted for exploration of posterior modes that correspond to plausible association hypotheses. JPT exhibits more accurate uncertainty representation of data associations with superior performance on standard metrics when compared to existing baselines. We also show the utility of JPT applied to automatic scheduling of user-in-the-loop annotations for improved trajectory quality.
... Elle est implémentée directement au niveau de l'hyperviseur, ou éventuellement au niveau d'une VM privilégiée qui a accès aux fonctions de l'hyperviseur. Cinq modules composent cette entité comme représenté dans la Figure 3. 3. Nous présentons l'ensemble de ces modules dans ce qui suit. ...
... Cependant, ce problème peut être plus compliqué si l'infrastructure Cloud adopte des topologies plus complexes telles que les topologies Jellyfish [85] ou Scafida [39]. Dans ce cas, ce problème est une instance du problème de partitionnement [3] et est NP-difficiles. Nous proposons donc une heuristique (cf. ...
... Le problème de répartition des VMs dans des clusters virtuels est une instance du problème d'optimisation de partitionnement d'un ensemble qui appartient à la classe des problèmes NP-difficiles [3]. Ainsi, à moins que = [30], il n'existe pas d'algorithmes qui s'exécutent en temps polynomial apportant une solution exacte à ce problème d'optimisation. ...
Since the emergence of Cloud Computing, the access to computing, storage and networking resources have never been so cheap and simple, which contributed to increase the hype around this paradigm. Nevertheless, there are still many challenges which have not been fully addressed in the Cloud. Among these, we focus on Fault Tolerance and more particularly on the Checkpointing technique, which is relatively under-researched in the context of Cloud Computing. To be used in Cloud Computing, this technique needs to be fully transparent and has to deal with the highly heterogeneous nature of the applications which are deployed in this environment.
We address this challenge in this thesis throughout our five contributions. First, we define a new architecture for the implementation of Fault Tolerance in Cloud Computing. Next, we address the evaluation needs of different Checkpointing approaches in the Cloud Computing by developing a new simulator. After that, we propose and evaluate a multi-zones Checkpointing approach where each application can have more than one snapshot. In the penultimate contribution, we propose and evaluate a new approach for the selection of the Checkpointing interval. Finally, in the last contribution, we present two new fully transparent and communication-aware Checkpointing approaches.
... (ii) ese hybrid metaheuristics are applied to the well-known set covering problem (SCP). is problem has been studied extensively in the literature, and therefore, there known instances where we can clearly evaluate the contribution of the db-scan binarization operator. On the other hand, the SCP has numerous practical real-world applications such as vehicle routing, railways, airline crew scheduling, microbial communities, and pattern finding [15][16][17][18]. (iii) Random operators are designed to study the contribution of the db-scan binarization algorithm in the binarization process. ...
... More recently, swarm-based metaheuristics, such as the cat swarm [28], cuckoo search [29], artificial bee colony [8], and black hole [30] metaheuristics, have also been proposed. e SCP has many practical applications in engineering, e.g., vehicle routing, railways, airline crew scheduling, microbial communities, and pattern finding [15,16,18,31]. e SCP can be formally defined as follows. Let A � (a ij ) be an n × m zero-one matrix, where a column j covers a row i if a ij � 1, and a column j is associated with a nonnegative real cost c j . ...
... (1) Function Transition (ListP(t), ListX(t), nClusters(t + 1)) (2) Input ListP(t), ListV i Clusters(t + 1), nClusters(t + 1) (3) Output List BinaryP(t + 1) (4) for x i (t), v x i (t + 1) in (ListP(t), ListV i (t + 1)) do (5) if v x i (t + 1) not in ouliers then (6) P tr (x i ) ⟵ getTransitionProbabily (ListV i Clusters(t + 1), nClusters(t + 1)) -equation (5) (7) else (8) P tr (x i ) ⟵ getOutlierTransitionProbabily (ListV i Clusters(t + 1), n Clusters(t + 1)) (9) end if (10) List BinaryP(t + 1).append (x i (t + 1)) ⟵ getBinaryPosition (P tr (x i (t)), ListV i Clusters(t + 1)) -equation (6) (11) end for (12) for x(t + 1) in List BinaryP(t + 1) do (13) List BinaryP(t + 1)[x(t + 1)] ⟵ Repair (x(t + 1)) (14) end for (15) return List BinaryP(t + 1) ALGORITHM 3: Transition algorithm. ...
The integration of machine learning techniques and metaheuristic algorithms is an area of interest due to the great potential for applications. In particular, using these hybrid techniques to solve combinatorial optimization problems (COPs) to improve the quality of the solutions and convergence times is of great interest in operations research. In this article, the db-scan unsupervised learning technique is explored with the goal of using it in the binarization process of continuous swarm intelligence metaheuristic algorithms. The contribution of the db-scan operator to the binarization process is analyzed systematically through the design of random operators. Additionally, the behavior of this algorithm is studied and compared with other binarization methods based on clusters and transfer functions (TFs). To verify the results, the well-known set covering problem is addressed, and a real-world problem is solved. The results show that the integration of the db-scan technique produces consistently better results in terms of computation time and quality of the solutions when compared with TFs and random operators. Furthermore, when it is compared with other clustering techniques, we see that it achieves significantly improved convergence times.
... The BDA problem is a binary integer linear program (BILP) with a search space complexity of O(2 N ×M + 2 M ), that is NP-hard [16] [17] with a smaller search space complexity compared to the DESAS problem. Turning off or on a BS impacts other network parameters such as available bandwidth in other cells or received interference. ...
... If utilization factor of BS j is above the threshold and if its associated macro BS has enough capacity (A mj ) to handle BS j load (line 6) we can turn off that base station and transfer its traffic load to its associated macro BS. If macro BS is overloaded, we can transfer BS j's load to one of its neighbors, if possible (lines [12][13][14][15][16][17][18][19]. However, if its neighbor cannot handle BS j's traffic load, we need to keep this BS on and use our power adaptation algorithm (PA()) that will be explained in the next section. ...
... Each base station independently monitors the feedback from active UEs. When the amount of received CQI value is higher than the threshold, the BS will reduce the power (P ) continuously until it makes sure the allocated power is higher than the minimum possible transmission power (P min ) and the new received CQI value is still equal or higher than the threshold (lines [15][16][17][18]. To enhance the network throughput, BSs need to allocate more power to the UEs located at the edges in comparison with other UEs. ...
Future mobile networks have to be densified by employing small cells to handle the upsurge in traffic load. Although the amount of energy each small cell consumes is low, the total energy consumption of a large-scale network may be enormous. To enhance energy efficiency, we have to adapt the number of active base stations to the offered traffic load. Deactivating base stations may cause coverage holes, degrade quality of service and throughput while redundant base stations waste energy. That is why we have to adapt the network to the effective density. In this paper, we show that achieving an optimal solution for adapting density of base stations to the demand is NP-hard. We propose a solution that consists of two heuristic algorithms: a base station density adaptation algorithm and a cell-zooming algorithm that determines which base stations must be kept active and adapts transmit power of base stations to enhance throughput, energy and spectral efficiency. We employ multi-access edge cloud for taking a snapshot of the network state in nearly real time with a wider perspective and for collecting network state over a large area. We show that the proposed algorithm conserves energy up to 12% while the spectral efficiency and network throughput can be enhanced up to 30% and 26% in comparison with recent works, respectively.
... That is why minimizing the non-covered demand locations or maximizing the covered demand locations is needed, and these properties all belong to the maximal covering problem (Laporte et al., 2015). In their study, Balas and Padberg (1976) argued that the set covering problem is one of the most applied models, along with the traveling salesman problem and set partitioning in integer programming (Balas & Padberg, 1976). In set covering problem, which covers a matrix of m rows and n columns with zero one ( ) matrix of a subset of the columns at minimum cost. ...
... That is why minimizing the non-covered demand locations or maximizing the covered demand locations is needed, and these properties all belong to the maximal covering problem (Laporte et al., 2015). In their study, Balas and Padberg (1976) argued that the set covering problem is one of the most applied models, along with the traveling salesman problem and set partitioning in integer programming (Balas & Padberg, 1976). In set covering problem, which covers a matrix of m rows and n columns with zero one ( ) matrix of a subset of the columns at minimum cost. ...
Route determination for perishable products is complex due to its unique characteristics, such as limited shelf-life regulatory requirements, or possibility of getting damaged. This research investigates a novel problem of collecting raw milk from a rural network of dairy farms. The research problem is grounded in a real scenario of milk collection in West Virginia, USA. The milk in this scenario is produced by small farms incapable of realizing transportation economies of density out in mostly rural areas throughout the state. Maximum coverage area and milk
processing overhead costs are used to identify suitable locations for intermediate milk collection centers or depots to store the milk to realize economies of density and reduce transportation costs. Each depot needs to be established within a certain maximum distance, for the refrigeration time of the multi-stop vehicle to not exceed the allowable time limit set to maintain the quality of the collected milk. This problem provides the unique opportunity to incorporate two separate classical optimization problems: the Set Covering Problem (SCP) (identifying depot locations) and theTraveling Salesman Problem (TSP) (routing). The SCP involves identifying the optimal number of service facilities required for the aggregation operations to maintain milk quality in transit. The TSP determines the most optimal route between farms and the depot. The milk collection problem requires solving both the SCP and TSP. However, the problem also becomes more difficult to solve when combining TSP constraints with the SCP. This study proposes a novel Mixed Integer Linear Programming (MILP) model to address this problem. The objective of the proposed model is to minimize the depot assignment cost, overhead cost of the depot, cleaning cost of the vehicle, and the vehicle distance traveled to reduce the fuel cost. The exact algorithm has been analyzed and we use sensitivity analyses to determine the model’s reliability and robustness to changes in the problem scenario. The model was tested for different scenarios for the dairy industry in West
Virginia. However, it has to be noted that other applications can be developed for similar structured problems based on this study given the flexible working path created by the Application Programming Interface (API) of Google Maps. We evaluate the proposed model by comparing its result with the steepest ascent Hill climbing algorithm, a mathematical optimization problem in Artificial Intelligence (AI) and Nearest Neighbor heuristics. The two algorithms are compared regarding solution quality and computational efficiency to determine the better heuristic algorithm for the developed model. The Hill climbing algorithm has given significantly better results than the Nearest Neighbor heuristics. The Hill climbing algorithm ended up in near-optimal results with an efficient computational time. Further, the Multi-Vehicle Routing (MVR) model is analyzed for the transportation part of the model and found that MVR scenario shows the potential over other
scenario (TSP-based) based on vehicle cycle time, still, there is a door to future research to incorporate the heterogeneous fleet and multi-depot constraints in the developed model.
... Eq. (51) ensures that each link in the physical topology is included in one and only one fiber tree of the FON. The optimization above (i.e., t-MILP) is equivalent to the problem of weighted set partitioning [48]. Although it is still an N P-hard problem, its formulation is compact and thus it can be solved quickly if the physical topology G(V, E) is not very large. ...
... Although it is still an N P-hard problem, its formulation is compact and thus it can be solved quickly if the physical topology G(V, E) is not very large. Meanwhile, there are a few existing approximation algorithms that can solve large-scale weighted set partitioning problems time-efficiently [48]. Therefore, we do not need to design an approximation algorithm for it here. ...
With enhanced cost-effectiveness, filterless optical networks (FONs) have been considered as a promising candidate for future optical infrastructure. However, as the transmission in FON relies on the "select-and-broadcast" scenario, it is more vulnerable to eavesdropping. Therefore, encrypting the communications in FONs will be indispensable, and this can be realized by introducing the optical transport network (OTN) encryption technologies that leverage high-speed encryption cards (ECs) to protect the integrity of OTN payload frames. In this paper, we study the problem of security-aware multilayer planning of FONs with OTN encryption. We first formulate a mixed integer linear programming (MILP) model (i.e., w-MILP) to solve the problem exactly. Then, to reduce the time complexity of problem-solving, we transform w-MILP into two correlated MILP models for establishing fiber trees for an FON (t-MILP) and planning flows in the fiber trees (s-MILP), respectively. The optimization in t-MILP is further transformed into a weighted set partitioning problem, which can be solved time-efficiently. As for s-MILP, we propose a polynomial-time approximation algorithm based on linear programming (LP) relaxation and randomized rounding. Extensive simulations verify the performance of our proposals.
... The Set Packing problem is a classical problem in pure Integer Programming and has been extensively studied (see Balas and Padberg (1976)). A Set Packing Problem (SP) can be formulated as an integer linear program of the form ...
... We introduce the connection between the SP problem and the Node Packing Problem, and explain how the study of the intersection graph of the latter is useful in the generation of valid inequalities for the former. A more in-depth introduction to these topics can be found in Padberg (1973); Balas and Padberg (1976); Atamtürk et al. (2000). ...
This doctorate is entirely devoted to an in-depth study of the Rank Pricing Problem (RPP) and two generalizations. The RPP is a combinatorial optimization problem which aims at setting the prices of a series of products of a company to maximize its revenue. This problem is specified by a set of unit-demand customers, that is, customers interested in a subset of the products offered by the company which intend to buy at most one of them. To do so, they count on a fixed budget, and they rank the products of their interest from the “best” to the “worst”. Once the prices are established by the company, they will purchase their highest-ranked product among the ones they can afford. In the RPP, it is assumed an unlimited supply of products, which is consistent with a company having enough copies of a product to satisfy the demand, or with a setting where the products can be produced quickly at negligible cost (e.g., digital goods). This dissertation consists of four chapters. The first chapter introduces the RPP problem and the mathematical concepts present in the work, whereas each of the next three chapters tackles the resolution of each of the problems of study: the RPP and two generalizations. Thus, Chapter 3 is dedicated to the Rank Pricing Problem with Ties (RPPT), an extension of the RPP where we consider that customers can express indifference among products in their preference list. And the last chapter of the thesis is devoted to a generalization of the problem that we have named the Capacitated Rank Pricing Problem (CRPP) with envy. For this generalization, we have considered reservation prices of customers for the different products that reflect their willingness to pay, instead of a single budget per customer. However, the main difference is that, in the CRPP, the company has a limited supply of products and might not be able to satisfy all the customers’ requests. This is a realistic assumption that we can find in many companies.The aim of this thesis is the proposal of mixed-integer linear formulations for the three problems of study, and their theoretical and/or computational comparison. The methodology used is based on the introduction of decision variables and adequate restrictions to model the problems. Another objective consists in strengthening the formulations by means of valid inequalities that reduce the feasible region of the relaxed problem and allow us to obtain better linear relaxation bounds. Finally, a third goal is to derive resolution algorithms for each of these models and compare them computationally, using commercial solvers.
... Definition 2. (Weighted set-partitioning problem [44]) Given a collection of sets S = {S 1 , S 2 , . . . , S l } where each set S i has a nonnegative weight w i and let F = ∪ l i=1 S i , the weighted setpartitioning problem is to find a subset of the family specified by I ⊆ {1, 2, . . . ...
... We show the NP-hardness of Problem 2 under a particular instance in which all the robots have the same feasible package groups as well as the same optimal cost to transport all the packages in each feasible package group. In Problem 2, each package group G, feasible for a robot k to serve while satisfying the corresponding time-windows, can be treated as a set in the weighted set-partitioning problem [44], and the corresponding total travel time c(k, G) for robot k to serve all the packages in G is the associated weight for the set. The goal of Problem 2 is to choose a set of such un-overlapping feasible package groups that each package group is specified for one certain robot (each specified robot is assigned exactly with one feasible package group), and the union of the chosen package groups contains all the packages to be delivered while minimizing the total cost for the robots to serve the feasible package groups. ...
This paper studies the multi-robot task assignment problem in which a fleet of dispersed robots needs to efficiently transport a set of dynamically appearing packages from their initial locations to corresponding destinations within prescribed time-windows. Each robot can carry multiple packages simultaneously within its capacity. Given a sufficiently large robot fleet, the objective is to minimize the robots' total travel time to transport the packages within their respective time-window constraints. The problem is shown to be NP-hard, and we design two group-based distributed auction algorithms to solve this task assignment problem. Guided by the auction algorithms, robots first distributively calculate feasible package groups that they can serve, and then communicate to find an assignment of package groups. We quantify the potential of the algorithms with respect to the number of employed robots and the capacity of the robots by considering the robots' total travel time to transport all packages. Simulation results show that the designed algorithms are competitive compared with an exact centralized Integer Linear Program representation solved with the commercial solver Gurobi, and superior to popular greedy algorithms and a heuristic distributed task allocation method.
... This static fog design is extended under the assumption of possible fog node failures to optimize the cost of design & dimensioning of dynamic fog nodes in the static fog infrastructure. This optimization is formulated as a set partitioning problem [14] via two approaches. 1) We propose an exact optimization approach (fog-RO-MILP) to the reliable design & dimensioning of a dynamic extension to our static fog infrastructure. ...
... Equations (22) and (24) are, respectively, the node association and path association constraints of the set partitioning problem [14]. Finally, the path association of a static path is bounded below by the appropriate fog node association. ...
Internet of Things (IoT) applications depend on reliable external storage and processing such as Cloud data centres. In response to high latency from Cloud, fog-computing has been introduced as a network of micro-data centres closer to IoT devices that provides a geo-distributed low-latency response. Current contributions regarding design & dimensioning of fog infrastructures are developed to service a static set of IoT traffic and a reliable fog network. However, these designs are not fault-tolerant. This article explores the implementation of reliable and fault-tolerant fog infrastructures via dynamically available fog nodes-standby nodes which activate when a nearby fog node fails. We formulate the design & dimensioning of dynamically available nodes as a set partitioning problem, which is solved via a mixed-integer linear program (MILP). This MILP formulation proves to be intractable; we therefore introduce a column generation approach to increase scalability with little loss to optimal design & dimensioning cost. Compared to other benchmark heuristic methods, our column generation approach yields reduced cost, with proportional solution time.
... Unlike the SRTA problem, which is an instance of the optimal assignment problem in combinatorial optimization, the MRTA problem, i.e., the instantaneous assignment problem (IAP), is an instance of the set partitioning problem (SPP) [37]. It comprises team division into non-overlapping subteams, also referred to as coalitions, that should conduct the tasks they are assigned to (multi-robot tasks in [5]). ...
... The complexity of task assignment and coalition formation is well investigated in the literature. For example, Balas and Padberg [37] analyzed the complexity of the related SPP, and Adams et al. [38] proved limits for the approximation of multi-robot coalition formation. Since we tackle dynamic missions and task assignment is performed in real-time, searching for the optimal solution to our MRTA problem is infeasible. ...
In many missions, multiple robots need to cooperate to complete tasks. The workflow of these multi-robot tasks involves forming coalitions of robots, assigning them to available tasks, and jointly executing the tasks. In this paper, we investigate such a workflow in an application-independent yet realistic setting. We abstract the key properties of tasks and robots and propose three distributed coalition formation and task assignment methods. Distributed methods rely on a timely and accurate exchange of state information in multi-robot systems (MRS). Thus we focus on two important communication aspects: (i) how to achieve consistent coalition formation and task assignment in the presence of communication faults, and (ii) how to reduce the communication effort required for the state updates in the MRS. In particular, we investigate the effect of event-triggered, time-triggered, and hybrid communication. We evaluate our distributed approaches in a simulation study using ns-3 and compare them with centralized methods and different network conditions. We demonstrate the sensitivity of complex missions to failure-prone MRS communication and provide robust, effective, and communication-aware methods for coalition formation and task assignment.
... The weighted set-partitioning problem (i.e., set cover with equality constraints) [19,3] is an optimization problem closely related to the exact cover problem. Given non-negative cost c S ≥ 0 for each S ∈ S, the problem asks for a minimum cost sub-collection of S that partitions X. ...
... This would already avert the design of an approximate separation oracle 4 to obtain a constant factor approximation for the LP relaxation of (P1) in the style of [8,24,18]. This would be in addition to the inherent difficulty with 3 Strictly speaking, we require T = 2 R so that maximizing over t ∈ T is the same as maximizing over t ⊆ R. This corresponds to the setting with lax quality of service constraints, which is what causes T to be exponential-sized to begin with. ...
The request-trip-vehicle assignment problem is at the heart of popular decomposition strategies for online vehicle routing. We study an integer linear programming formulation and its linear programming relaxation. Our main result is a simple, linear programming based randomized algorithm that, whenever the instance is feasible, leverages assumptions typically met in practice to return an assignment whose: i) expected cost is at most that of an optimal solution, and ii) expected fraction of unassigned requests is at most $1/e$. If trip-vehicle assignment costs can only be $\alpha$-approximated, we pay an additional factor of $\alpha$ in the expected cost. Unassigned requests are assigned in future rounds with high probability. We can relax the feasibility requirement by including a penalty term for unassigned requests, in which case our performance guarantee is with respect to a modified objective function. Our techniques generalize to a class of set-partitioning problems.
... Finally, the set partitioning problem is to find a minimum cost partition of M , i.e. a collection F ⊆ N which is both a cover and a packing, {min c T x : Ax = e, x ∈ {0, 1} n }. A comprehensive survey on theory and applications of these three models is presented for example in Balas and Padberg (1976) and Vemuganti (1998). The set partitioning, covering and packing models are strictly related. ...
... Indeed, set partitioning can be brought to set covering and packing. Furthermore, set packing can be restated as a set partitioning (see Balas and Padberg (1976)). The three problems can be combined in a unified model, which aims to allow under (over) coverage, yielding set packing (covering). ...
Personnel scheduling problems encompass a large collection of optimization challenges thatseveral organizations need to face. One of the first classifications proposed divides theseproblems into: days-o_ scheduling, shift scheduling, and tour scheduling. The first concernsthe determination of working and rest days; the second defines the shifts to be assigned; thethird integrates days-o_ into shift scheduling. The current thesis addresses the tour scheduling problem with a multi-activity context arising in restaurant business, which was provided by the company Horizontal Software. As such, the main goal is to define the working days and the shifts, along with the specification of the activities in each time period. This problem is also characterized by a high degree of flexibility, mainly due to the introduction of a long pause (interruption), and heterogeneity, determined by conditions from employees (skills, availabilities, contract regulations, and pre-assignments). The problem is tackled by a Branch-and-Price algorithm, which is based on column generation. A dual ascent heuristic is implemented to speed up its convergence, and a constraint and dynamic programming based method is proposed for generating schedules. To address the large-scale instances, various heuristics are presented, based on column generation, large neighborhood search and tabu search. To assess the performance of the proposed method,computational experiments are conducted on instances from the literature and on real-worldinstances that were provided by Horizontal Software.
... In C-RAN, mapping RRH clusters to BBU pools is not trivial, since the clusters also need to meet some explicit and implicit constraints, including the geographic distance of the cluster, the global constraints on the resource blocks available, etc [5]. Such a problem has been identified as set partitioning problem [48], [49], and its complexity is proven to be NP-hard [50]. Therefore, exhaustively searching for every possible mapping scheme is computationally intractable as the network scale increases [8]. ...
... In this phase, given the RRH traffic and handover predictions as well as the BBU pool constraints, our objective is to design an optimal RRH-BBU mapping scheme that maximizes BBU utilization rate and minimizes RRH handover overhead. Such a problem has been identified as set partitioning problem [48], [49], and its complexity is proven to be NP-hard [50]. Therefore, exhaustively searching for every possible mapping scheme is computationally intractable as the network scale increases [8]. ...
The surging traffic volumes and dynamic user mobility patterns pose great challenges for cellular network operators to reduce operational costs and ensure service quality. Cloud Radio Access Network (C-RAN) aims to address these issues by handling traffic and mobility in a centralized manner, separating baseband units (BBUs) from base stations (RRHs) and sharing BBUs in a pool. The key problem in C-RAN optimization is to dynamically allocate BBUs and map them to RRHs under cost and quality constraints, since real-world traffic and mobility are difficult to predict, and there are enormous numbers of candidate RRH-BBU mapping schemes. In this work, we propose a data-driven framework for C-RAN optimization. Firstly, we propose a deep-learning-based Multivariate Long Short Term Memory (MuLSTM) model to capture the spatiotemporal patterns of traffic and mobility for accurate prediction. Secondly, we formulate RRH-BBU mapping with cost and quality objectives as a set partitioning problem, and propose a Resource-Constrained Label-Propagation (RCLP) algorithm to solve it. We show that the greedy RCLP algorithm is monotone suboptimal with worst-case approximation guarantee to optimal. Evaluations with real-world datasets from Ivory Coast and Senegal show that our framework achieves a BBU utilization above 85.2%, with over 82.3% of mobility events handled with high quality, outperforming the traditional and the state-of-the-art baselines.
... The approach to the problem under consideration we propose is based on the idea to generate a set of feasible groups, each one associated to a specific route, obtained by considering all (or almost all) possible combinations of users under a set of specific conditions. Then, an optimization model implementing a multi-objective set-partitioning approach (see Balas and Padberg 1976;Hoffman et al. 2009) is solved. ...
The number of people using shared or smart mobility for daily travel and for tourism purposes is growing. This work aims to define a new smart decision approach to promotes the use of car-pooling by tourist groups to reach sites of interest that are difficult to reach with other public transport solutions. The decision problem consists in defining a set of car groups, each of which is associated with a specific route, obtained by considering all (or nearly all) the possible combinations, according to specific social rating requirements. We propose a new two-stage intelligent decision-making approach, based on the set partitioning model, with a path generation procedure and a multi-objective approach. Several computational experiments have been carried out in order to validate the effectiveness of the proposed approach and the impact of the weights assigned to the different optimization criteria and of the social requirements on the overall car groups definition. The solutions obtained show a benefit compared to the initial situation, under each of the performance measures adopted. We have also shown how the introduction of conditions on the social rating affects the efficiency of the solutions and that all the solutions analyzed above are Pareto-optimal and represent the best planning of the automotive groups according to different attitudes regarding the optimization criteria. We have also shown how the introduction of conditions on social rating affects the efficiency of the solutions.
... 1 1 360 395 0 1 2 1 410 455 1 2 3 1 460 502 2 1 4 1 508 540 1 0 Table 1: A Bus Tour Example Branch and Price is a decomposition technique for large mixed integer programs (Barnhart et al. 1998). This work uses set partitioning (Balas and Padberg 1976) as the master problem and the RCSPP (Irnich and Desaulniers 2005) as the subproblem. Resources are modelled via resource extension functions (REF) (Irnich 2008). ...
This paper presents a Branch and Price approach for a real-life Bus Driver Scheduling problem with a complex set of break constraints. The column generation uses a set partitioning model as master problem and a resource constrained shortest path problem as subproblem. Due to the complex constraints, the branch and price algorithm adopts several novel ideas to improve the column generation in the presence of a high-dimensional subproblem, including exponential arc throttling and a dedicated two-stage dominance algorithm. Evaluation on a publicly available set of benchmark instances shows that the approach provides the first provably optimal solutions for small instances, improving best-known solutions or proving them optimal for 48 out of 50 instances, and yielding an optimality gap of less than 1% for more than half the instances.
... It is not easy to discover the optimal partition . This problem is an instance of the set partitioning problem [10], which is known to be NP-complete, where the objective function is computed by brute-force searching for every combination of candidate blocks. It is hard to solve since the search space is basically a very large scale due to large | ( )|. ...
How can we explore the unknown properties of high-dimensional sensitive relational data while preserving privacy? We study how to construct an explorable privacy-preserving materialized view under differential privacy. No existing state-of-the-art methods simultaneously satisfy the following essential properties in data exploration: workload independence, analytical reliability (i.e., providing error bound for each search query), applicability to high-dimensional data, and space efficiency. To solve the above issues, we propose HDPView, which creates a differentially private materialized view by well-designed recursive bisected partitioning on an original data cube, i.e., count tensor. Our method searches for block partitioning to minimize the error for the counting query, in addition to randomizing the convergence, by choosing the effective cutting points in a differentially private way, resulting in a less noisy and compact view. Furthermore, we ensure formal privacy guarantee and analytical reliability by providing the error bound for arbitrary counting queries on the materialized views. HDPView has the following desirable properties: (a) Workload independence, (b) Analytical reliability, (c) Noise resistance on high-dimensional data, (d) Space efficiency. To demonstrate the above properties and the suitability for data exploration, we conduct extensive experiments with eight types of range counting queries on eight real datasets. HDPView outperforms the state-of-the-art methods in these evaluations.
... Set partitioning problem is well known to be NP-hard [40]. Consequently, the MAP being a special set partitioning problem is also NP-hard. ...
The term “green networking” refers to energy-efficient networking technologies and products, while minimizing resource usage as possible. This thesis targets the problem of resource allocation in small cells networks in a green networking context. We develop algorithms for different paradigms. We exploit the framework of coalitional games theory and some stochastic geometric tools as well as the crowding game model. We first study the mobile assignment problem in broadcast transmission where minimal total power consumption is sought. A green-aware approach is followed in our algorithms. We examine the coalitional game aspects of the mobile assignment problem. This game has an incentive to form grand coalition where all players join to the game. By using Bondareva-Shapley theorem, we prove that this coalitional game has a non-empty core which means that the grand coalition is stable. Then, we examine the cost allocation policy for different methods. In a second part, we analyze a significant problem in green networking called switching off base stations in case of cooperating service providers by means of stochastic geometric and coalitional game tools. The coalitional game herein considered is played by service providers who cooperate in switching off base stations. We observed the Nash stability which is a concept in hedonic coalition formation games. We ask the following question: Is there any utility allocation method which could result in a Nash-stable partition? We address this issue in the thesis. We propose the definition of the Nash-stable core which is the set of all possible utility allocation methods resulting in stable partitions obtained according to Nash stability. We finally consider games related to the association of mobiles to an access point. The player is the mobile which has to decide to which access point to connect. We consider the choice between two access points or more, where the access decisions may depend on the number of mobiles connected to each access points. We obtained new results using elementary tools from congestion and crowding games. Last but not least, we extend our work to cooperative transmissions. We formulate the partner selection problem in cooperative relaying based on a matching theoretic approach. Partner selection is described as a special stable roommate problem where each player ranks its partners by some criterion. We adapted Irving’s algorithm for determining the partner of each player. We introduced a decentralized version of the Irving’s algorithm.
... There is a wide variety of optimization approaches available, and the use of these approaches depends on the nature and the degree of complexity of the problem to be optimized (Lenagh 2013). Deterministic techniques include numerical and classical methods such as graphical methods, gradient, and Hessian based methods, derivative-free approaches, quadratic programming, sequential quadratic programming, penalty methods, etc. (Balas and Padberg 1976). In Atay and Bayazit (2006), a mixed-integer linear programming optimization approach was used in order to allocate heterogeneous robots for maximizing the coverage area of the regions of interest. ...
In this paper, a new multi-robot task allocation (MRTA) algorithm inspired by the Newtonian law of gravity is proposed. In the proposed method, targets and robots are considered as fixed objects and movable objects, respectively. For each target, a constant mass is assigned, which corresponds to its quality. The fixed objects (which refer to targets) apply a gravitational force to the movable objects (which are considered as robots) and change their positions in the feasible search space and therefore, the best target allocation of robots is determined by employing the law of gravity. In the proposed scenario, task allocation consists of assigning the robots to the found targets in a 2-D feasible area. The expected distribution is obtained from the targets’ qualities that are represented as scalar values. Decision-making is a distributed mechanism and robots choose their assignments, taking into account targets’ qualities and distances. Moreover, a control parameter is planned to make a remarkable balance between exploration and exploitation ability of the proposed algorithm. A self-adaptive mechanism is proposed to adjust the value of the exploration parameter automatically, aiming to maintain the balance between exploration and exploitation ability of robots. Furthermore, in order to decrease the time of reaching the target and accelerate computation, a selection memory is designed. In the experiments, we examine the scalability of the proposed method in terms of the number of robots and the number of targets and speed of algorithm to deliver robots to the desired targets with comparison to other competitors. The simulation results show the scalability of the algorithm, comparing the existing methods. Moreover, some non-parametric statistical tests are utilized to compare the results obtained in experiments. The statistical comparisons confirm the superiority of the proposed method compared over the existing methods.
... Another prospective research topic is to develop an optimization scheme to solve the minimization problem in (10). In fact, this problem is a type of set partitioning problem (see [27], [28] for background material) on the sample setŜ 1 . Hence, applying some advanced knowledge about the set partitioning problem to this minimization problem would be interesting. ...
Shape matching with local descriptors is an underlying scheme in shape analysis. We can visually confirm the matching results and also assess them for shape classification. Generally, shape matching is implemented by determining the correspondence between shapes that are represented by their respective sets of sampled points. Some matching methods have already been proposed; the main difference between them lies in their choice of matching cost function. This function measures the dissimilarity between the local distribution of sampled points around a focusing point of one shape and the local distribution of sampled points around a referring point of another shape. A local descriptor is used to describe the distribution of sampled points around the point of the shape. In this paper, we propose an extended scheme for shape matching that can compensate for errors in existing local descriptors. It is convenient for local descriptors to adopt our scheme because it does not require the local descriptors to be modified. The main idea of our scheme is to consider the correspondence of neighboring sampled points to a focusing point when determining the correspondence of the focusing point. This is useful because it increases the chance of finding a suitable correspondence. However, considering the correspondence of neighboring points causes a problem regarding computational feasibility, because there is a substantial increase in the number of possible correspondences that need to be considered in shape matching. We solve this problem using a branch-and-bound algorithm, for efficient approximation. Using several shape datasets, we demonstrate that our scheme yields a more suitable matching than the conventional scheme that does not consider the correspondence of neighboring sampled points, even though our scheme requires only a small increase in execution time.
... The classical set covering combinatorial problem (SCP) is an important NP-hard problem, which has not only theoretical importance in the field of optimization but also from a practical point of view as it has important practical applications in different areas of engineering, for example, in the vehicle routing, railroads, airline crew scheduling, microbial communities, and pattern finding [57][58][59][60]. ...
The optimization methods and, in particular, metaheuristics must be constantly improved to reduce execution times, improve the results, and thus be able to address broader instances. In particular, addressing combinatorial optimization problems is critical in the areas of operational research and engineering. In this work, a perturbation operator is proposed which uses the k-nearest neighbors technique, and this is studied with the aim of improving the diversification and intensification properties of metaheuristic algorithms in their binary version. Random operators are designed to study the contribution of the perturbation operator. To verify the proposal, large instances of the well-known set covering problem are studied. Box plots, convergence charts, and the Wilcoxon statistical test are used to determine the operator contribution. Furthermore, a comparison is made using metaheuristic techniques that use general binarization mechanisms such as transfer functions or db-scan as binarization methods. The results obtained indicate that the KNN perturbation operator improves significantly the results.
... We now provide a very brief summary of the set-packing polytope and some of its properties-we direct the reader to [2] for a comprehensive review. ...
We present an integer programming model to compute the strong rainbow connection number $src(G)$ of any simple graph $G$. We introduce several enhancements to the proposed model, including a fast heuristic, a novel class of valid inequalities, and a variable elimination scheme. Moreover, we present a novel lower bound for $src(G)$ which may be of independent research interest. We evaluate our model with a traditional branch and cut approach as well as an alternative scheme based on iterative lower bound improvement, which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs with up to $167$ vertices.
... In this phase, given the predicted RRH traffic and handover in a future period of time, as well as the BBU size and pool capacity constraints, our objective is to find optimal RRH-BBU mapping scheme to maximize BBU utilization rate and minimize RRH handover overhead. Such a problem has been identified as set partitioning problem [195,196], and its complexity is proven to be NP-hard [197]. Therefore, exhaustively searching for every possible mapping scheme is computationally intractable as the network scale increases [28]. ...
The evolution of metropolitan structures and the development of urban systems have created various kinds of urban networks, among which two types of networks are of great importance for our daily life, the transportation networks corresponding to human mobility in the physical space, and the communication networks supporting human interactions in the digital space. The rapid expansion in the scope and scale of these two networks raises a series of fundamental research questions on how to optimize these networks for their users. Some of the major objectives include demand responsiveness, anomaly awareness, cost effectiveness, energy efficiency, and service quality. Despite the distinct design intentions and implementation technologies, both the transportation and communication networks share common fundamental structures, and exhibit similar spatio-temporal dynamics. Correspondingly, there exists an array of key challenges that are common in the optimization in both networks, including network profiling, mobility prediction, traffic clustering, and resource allocation. To achieve the optimization objectives and address the research challenges, various analytical models, optimization algorithms, and simulation systems have been proposed and extensively studied across multiple disciplines. Generally, these simulation-based models are not evaluated in real-world networks, which may lead to sub-optimal results in deployment. With the emergence of ubiquitous sensing, communication and computing diagrams, a massive number of urban network data can be collected. Recent advances in big data analytics techniques have provided researchers great potentials to understand these data. Motivated by this trend, we aim to explore a new big data-driven network optimization paradigm, in which we address the above-mentioned research challenges by applying state-of-the-art data analytics methods to achieve network optimization goals. Following this research direction, in this dissertation, we propose two data-driven algorithms for network traffic clustering and user mobility prediction, and apply these algorithms to real-world optimization tasks in the transportation and communication networks. First, by analyzing large-scale traffic datasets from both networks, we propose a graph-based traffic clustering algorithm to better understand the traffic similarities and variations across different area and time. Upon this basis, we apply the traffic clustering algorithm to the following two network optimization applications. 1. Dynamic traffic clustering for demand-responsive bikeshare networks. In this application, we dynamically cluster bike stations with similar usage patterns to obtain stable and predictable cluster-wise bike traffic demands, so as to foresee over-demand stations in the network and enable demand-responsive bike scheduling. Evaluation results using real-world data from New York City and Washington, D.C. show that our framework accurately foresees over-demand clusters (e.g. with 0.882 precision and 0.938 recall in NYC), and outperforms other baseline methods significantly. 2. Complementary traffic clustering for cost-effective C-RAN. In this application, we cluster RRHs with complementary traffic patterns (e.g., an RRH in residential area and an RRH in business district) to reuse the total capacity of the BBUs, so as to reduce the overall deployment cost. We evaluate our framework with real-world network data collected from the city of Milan, Italy and the province of Trentino, Italy. Results show that our method effectively reduces the overall deployment cost to 48.4\% and 51.7\% of the traditional RAN architecture in the two datasets, respectively, and consistently outperforms other baseline methods. Second, by analyzing large-scale user mobility datasets from both networks, we propose [...]
... In this section we alternatively reformulate (MOMFHP 0 ) as a set partitioning problem (SPP) (see e.g., Balas and Padberg (1976)). Our SPP is based on the idea that once the p clusters of demand points are known , (MOMFHP 0 ) reduces to finding the optimal hyperplanes for each of those clusters in which all the residuals are aggregated by means of an ordered median function. ...
In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances between points and hyperplanes are aggregated by means of ordered median functions. A compact Mixed Integer Linear (or Non Linear) programming formulation is presented for the problem and also an extended set partitioning formulation with an exponential number of variables is derived. We develop a column generation procedure embedded within a branch-and-price algorithm for solving the problem by adequately performing its preprocessing, pricing and branching. We also analyze geometrically the optimal solutions of the problem, deriving properties which are exploited to generate initial solutions for the proposed algorithms. Finally, the results of an extensive computational experience are reported. The issue of scalability is also addressed showing theoretical upper bounds on the errors assumed by replacing the original datasets by aggregated versions.
... Optimization in Data Association. Data association can be interpreted as the Set Partition Problem (SPP) [4]. In multi-object tracking, this SPP formalism is specialized into the Multi-Dimensional Assignment (MDA) problem, which describes the optimization procedure defined on a k-partite graph to partition object observations into trajectories cross k frames. ...
In this work, we present an end-to-end framework to settle data association in online Multiple-Object Tracking (MOT). Given detection responses, we formulate the frame-by-frame data association as Maximum Weighted Bipartite Matching problem, whose solution is learned using a neural network. The network incorporates an affinity learning module, wherein both appearance and motion cues are investigated to encode object feature representation and compute pairwise affinities. Employing the computed affinities as edge weights, the following matching problem on a bipartite graph is resolved by the optimization module, which leverages a graph neural network to adapt with the varying cardinalities of the association problem and solve the combinatorial hardness with favorable scalability and compatibility. To facilitate effective training of the proposed tracking network, we design a multi-level matrix loss in conjunction with the assembled supervision methodology. Being trained end-to-end, all modules in the tracker can co-adapt and co-operate collaboratively, resulting in improved model adaptiveness and less parameter-tuning efforts. Experiment results on the MOT benchmarks demonstrate the efficacy of the proposed approach.
... Clique Cuts. The clique inequalities are well-known valid inequalities for the set partitioning problem (Balas and Padberg 1976), which induce facets. They are defined over an undirected conflict graph G (Ω, E ), in which an edge between two vertices r 1 , r 2 ∈ Ω exists if r 1 and r 2 both visit a customer i ∈ V (i.e., a r 1 i ≥ 1 and a r 2 i ≥ 1). ...
Vehicle routing problems (VRPs) are among the most studied problems in operations research. Nowadays, the leading exact algorithms for solving many classes of VRPs are branch-price-and-cut algorithms. In this survey paper, we highlight the main methodological and modeling contributions made over the years on branch-and-price (branch-price-and-cut) algorithms for VRPs, whether they are generic or specific to a VRP variant. We focus on problems related to the classical VRP—that is, problems in which customers must be served by several capacitated trucks and which are not combinations of a VRP and another optimization problem.
... Given all the possible , ′ calculated in Stage 1, the process of minimizing formula (6) becomes a set partitioning problem (Balas and Padberg 1976): find p number of , ′ with the smallest summation, subject to the constraints that all the geographic units are included and each unit can be included in only one map class. This set partitioning problem can be formulated as follows: ...
Choropleth mapping provides a powerful way to visualize geographical phenomena with colors, shadings, or patterns. In many real-world applications, geographical data often contain uncertainty. How to incorporate such uncertainty into choropleth mapping is challenging. Although a few existing methods attempt to address the uncertainty issue in choropleth mapping, there are limitations to widely applying these methods due to their strong assumption on the distribution of uncertainty and the way in which similarity or dissimilarity is assessed. This article provides a new classification scheme for choropleth maps when data contain uncertainty. Considering that in a choropleth map, units in the same class are assigned with the same color or pattern, this new approach assumes the existence of a representative value for each class. A maximum likelihood estimation–based approach is developed to determine class breaks so that the overall within-class deviation is minimized while considering uncertainty. Different methods—including linear programming, dynamic programming, and an interchange heuristic—are developed to solve the new classification problem. The proposed mapping approach has been applied to map the median household income data from the American Community Survey and simulated disease occurrence data. Test results show the effectiveness of the new approach. The linkage between the new approach and the existing methods is also discussed. Key Words: choropleth mapping, map classification, maximum likelihood estimation, uncertainty.
Set cover problems are among the most popular and well-studied models in facility location. In this paper, we address an extension of the set covering and partial set covering problems. We suppose that a set of customers to be served is divided into two non-overlapped subsets: “mandatory” customers who must always be covered and “optional” customers who do not require obligatory coverage. To avoid a trivial solution that covers only customers from H, we suppose that the number of covered optional customers must be at least larger than a pre-defined threshold In the real world, local laws may prohibit location of a large number of facilities in some places to reduce harm to people and/or nature. To reflect that fact our problem involves pairs of facilities that are mutually exclusive, hence if a facility from such a pair is located, another one is prohibited. We formulate this problem as an integer linear program and develop and implement several fast heuristic approaches capable of finding high-quality solutions of large-scale problem instances arisen in real-life settings. In particular, we develop a hybrid heuristic involving a greedy algorithm and local search embedded into a variable neighborhood framework. We test our algorithms in a series of intensive computational experiments on both real-life problems and synthetically generated instances.
Machine learning algorithms have been increasingly integrated into applications that significantly affect human lives. This surged an interest in designing algorithms that train machine learning models to minimize training error and imposing a certain level of fairness. In this paper, we consider the problem of fair clustering of data sets. In particular, given a set of items each associated with a vector of nonsensitive attribute values and a categorical sensitive attribute (e.g., gender, race, etc.), our goal is to find a clustering of the items that minimizes the loss (i.e., clustering objective) function and imposes fairness measured by Rényi correlation. We propose an efficient and scalable in-processing algorithm, driven by findings from the field of combinatorial optimization, that heuristically solves the underlying optimization problem and allows for regulating the trade-off between clustering quality and fairness. The approach does not restrict the analysis to a specific loss function, but instead considers a more general form that satisfies certain desirable properties. This broadens the scope of the algorithm’s applicability. We demonstrate the effectiveness of the algorithm for the specific case of k-means clustering as it is one of the most extensively studied and widely adopted clustering schemes. Our numerical experiments reveal the proposed algorithm significantly outperforms existing methods by providing a more effective mechanism to regulate the trade-off between loss and fairness.
History: Rema Padman served as the senior editor for this article.
Data Ethics & Reproducibility Note: The code capsule is available on Code Ocean at https://codeocean.com/capsule/9556728/tree and in the e-Companion to the this article (available at https://doi.org/10.1287/ijds.2022.0005 ).
Problem definition: Infectious disease screening can be expensive and capacity constrained. We develop cost- and capacity-efficient testing designs for multidisease screening, considering (1) multiplexing (disease bundling), where one assay detects multiple diseases using the same specimen (e.g., nasal swabs, blood), and (2) pooling (specimen bundling), where one assay is used on specimens from multiple subjects bundled in a testing pool. A testing design specifies an assay portfolio (mix of single-disease/multiplex assays) and a testing method (pooling/individual testing per assay). Methodology/results: We develop novel models for the nonlinear, combinatorial multidisease testing design problem: a deterministic model and a distribution-free, robust variation, which both generate Pareto frontiers for cost- and capacity-efficient designs. We characterize structural properties of optimal designs, formulate the deterministic counterpart of the robust model, and conduct a case study of respiratory diseases (including coronavirus disease 2019) with overlapping clinical presentation. Managerial implications: Key drivers of optimal designs include the assay cost function, the tester’s preference toward cost versus capacity efficiency, prevalence/coinfection rates, and for the robust model, prevalence uncertainty. When an optimal design uses multiple assays, it does so in conjunction with pooling, and it uses individual testing for at most one assay. Although prevalence uncertainty can be a design hurdle, especially for emerging or seasonal diseases, the integration of multiplexing and pooling, and the ordered partition property of optimal designs (under certain coinfection structures) serve to make the design more structurally robust to uncertainty. The robust model further increases robustness, and it is also practical as it needs only an uncertainty set around each disease prevalence. Our Pareto designs demonstrate the cost versus capacity trade-off and show that multiplexing-only or pooling-only designs need not be on the Pareto frontier. Our case study illustrates the benefits of optimally integrated designs over current practices and indicates a low price of robustness.
Funding: This work was supported by the National Science Foundation [Grant 1761842].
Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0296 .
This article considers an optimization problem aiming at planning the service stations to recharge a fleet of electric buses (EBs) for public transportation. The planning problem includes the selection of the sites of the stations (among a set of pre-specified eligible sites) and their sizing (both in terms of the number of sockets and maximum output power). The optimization problem also integrates issues related to the assignment of the various lines to the activated stations and the bus fleet's sizing. The formalization of the optimization problem is carried out by taking into account the nonlinear charging process characteristic of the batteries of EBs. The optimization problem is defined as a minimum cost design while guaranteeing a minimum quality level of the service, represented in this case by a minimum frequency for each of the lines. The overall decision problem turns out to be a mixed nonlinear one. It can be efficiently solved by common commercial software tools for moderate-medium problem sizes, as demonstrated in the presented application.
This work revisits novel variants of the Simple Plant Location Problem (SPLP), which were originally presented in the PhD thesis Set packing, location and related problems (2019). They consist in considering different requirements on clients allocation to facilities, namely incompatibilities or co-location for some specific pairs of clients. These correspond to “cannot-link” and “must-link” constraints previously studied in the related topic of clustering. Set packing formulations of these new location problems are presented, together with families of valid inequalities and facets. Computational experience supporting the effectiveness of the resulting cuts into a branch and cut procedure is exposed. The comprehensive revision on these new variants of the SPLP is completed by illustrative applications, including the design of telecommunication networks and interesting connections in the field of Genetics, and also by the analysis of open questions and future research directions.
How can we explore the unknown properties of high-dimensional sensitive relational data while preserving privacy? We study how to construct an explorable privacy-preserving materialized view under differential privacy. No existing state-of-the-art methods simultaneously satisfy the following essential properties in data exploration: workload independence, analytical reliability (i.e., providing error bound for each search query), applicability to high-dimensional data, and space efficiency. To solve the above issues, we propose HDPView, which creates a differentially private materialized view by well-designed recursive bisected partitioning on an original data cube, i.e., count tensor. Our method searches for block partitioning to minimize the error for the counting query, in addition to randomizing the convergence, by choosing the effective cutting points in a differentially private way, resulting in a less noisy and compact view. Furthermore, we ensure formal privacy guarantee and analytical reliability by providing the error bound for arbitrary counting queries on the materialized views. HDPView has the following desirable properties: (a) Workload independence , (b) Analytical reliability , (c) Noise resistance on high-dimensional data , (d) Space efficiency. To demonstrate the above properties and the suitability for data exploration, we conduct extensive experiments with eight types of range counting queries on eight real datasets. HDPView outperforms the state-of-the-art methods in these evaluations.
The problem of representing and querying sensor network readings issues new research challenges, as traditional techniques and architectures used for managing relational and object oriented databases are not suitable in this context. In this paper, we present a Grid-based framework that supports aggregate query answering on sensor network data, and uses a summarization technique to efficiently accomplish this task. In particular, Grid nodes are used for collecting, compressing and storing sensor network readings, as well as extracting information from stored data. Grid nodes can exchange information among each other, so that the same piece of information can be stored (with a different degree of accuracy) into several nodes. Queries are evaluated by locating the Grid nodes containing the needed information (either compressed or not), and choosing (among these nodes) the most convenient ones, according to a cost model. We complete our contribution with a case study that focuses the attention on the management and querying of Grid-based GIS databases.
This letter considersa particular class of multi-robot task allocation problems, where tasks correspond to heterogeneous multi-robot routing problems defined on different areas of a given environment. We present a hierarchical planner that breaks down the complexity of this problem into two subproblems: the high-level problem of allocating robots to routing tasks, and the low-level problem of computing the actual routing paths for each subteam. The planner uses a Graph Neural Network (GNN) as a heuristic to estimate subteam performance for specific coalitions on specific routing tasks. It then iteratively refines the estimates to the real subteam performances as solutions of the low-level problems become availableon a testbed problem having a heterogeneous multi-robot area inspection problem as the base routing task, we empirically show that our hierarchical planner is able to compute optimal or near-optimal (within 7%) solutions approximately 16 times faster (on average) than an optimal baseline that computes plans for all the possible allocations in advance to obtain precise routing times. Furthermore, we show that a GNN-based estimator can provide an excellent trade-off between solution quality and computation time compared to other baseline (non-learned) estimators.
In this paper, we consider the P-prize-collecting set cover (P-PCSC) problem, which is a generalization of the set cover problem. In this problem, we are given a set system \((U, {\mathscr {S}})\), where U is a ground set and \({\mathscr {S}}\subseteq 2^{U}\) is a collection of subsets satisfying \(\cup _{S\in {\mathscr {S}}}S=U\). Every subset in \({\mathscr {S}}\) has a nonnegative cost, and every element in U has a nonnegative penalty cost and a nonnegative profit. Our goal is to find a subcollection \({\mathscr {C}}\subseteq {\mathscr {S}}\) such that the total cost, consisting of the cost of subsets in \({\mathscr {C}}\) and the penalty cost of the elements not covered by \({\mathscr {C}}\), is minimized and at the same time the combined profit of the elements covered by \({\mathscr {C}}\) is at least P, a specified profit bound. Our main work is to obtain a \(2f+\varepsilon \)-approximation algorithm for the P-PCSC problem by using the primal-dual and Lagrangian relaxation methods, where f is the maximum frequency of an element in the given set system \((U, {\mathscr {S}})\) and \(\varepsilon \) is a fixed positive number.
The surging data traffic and dynamic user mobility in 5G networks have posed significant demands for mobile operators to increase data processing capacity and ensure user handover quality. Specifically, a cost-effective and quality-aware radio access network (RAN) is in great necessity. With the emergence of fog-computing-based RAN architecture (Fog-RAN), the data processing units (BBUs) can be separated from base stations (RRHs) and hosted in distributed fog servers, where each server accommodates a community of RRHs to handle data traffic and user handover. The key problem in Fog-RAN optimization is how to cluster complementary RRHs into communities and allocate adequate numbers of BBUs for the fog servers, since real-world traffic and mobility patterns are highly dynamic to model, and it is not trivial to find an optimal RRH clustering and BBU allocation scheme from potentially enormous numbers of candidates. In this work, we propose a data-driven framework for cost-effective and quality-aware Fog-RAN optimization. In the RRH clustering phase, we build a weighted graph model to characterize user mobility patterns across RRHs, and propose a size-constrained community detection (SCUD) algorithm to cluster RRHs into communities with frequent internal handover events. In the BBU allocation phase, we formulate BBU allocation in each community fog server as a set partitioning problem, and propose a column-reduced integer programming (CLIP) algorithm to find optimal BBU allocation schemes that maximize BBU utilization rate. Evaluations using two large-scale real-world datasets collected from Ivory Coast and Senegal show that compared to the traditional RAN architecture, our framework effectively reduces the average handover overhead to 12.8% and 27.3%, and increases the average BBU utilization rate to 49.7% and 52.3% in both cities, respectively, which consistently outperforms the state-of-the-art baseline methods.
In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances between points and hyperplanes are aggregated by means of ordered median functions. A compact Mixed Integer Linear (or Non Linear) programming formulation is presented for the problem and also an extended set partitioning formulation with a huge number of variables is derived. We develop a column generation procedure embedded within a branch-and-price algorithm for solving the problem by adequately performing its preprocessing, pricing and branching. We also analyze geometrically the optimal solutions of the problem, deriving properties which are exploited to generate initial solutions for the proposed algorithms. Finally, the results of an extensive computational experience are reported. The issue of scalability is also addressed showing theoretical upper bounds on the errors assumed by replacing the original datasets by aggregated versions.
When deciding where to locate facilities (e.g., emergency points where an ambulance will wait for a call) that provide a service, it happens quite often that a customer (e.g., a person) can receive this service only if she is located less than a certain distance from the nearest facility (e.g., the ambulance can arrive in less than 7 min at this person’s home). The problems that share this property receive the name of covering problems and have many applications. (analysis of markets, archaeology, crew scheduling, emergency services, metallurgy, nature reserve selection, etc.). This chapter surveys the most relevant problems in this field: the Set Covering Problem, the Maximal Covering Location Problem, and related problems, In addition, it is introduced a general model that has as particular cases the main covering location models. The most important theoretical results in this topic as well as exact and heuristic algorithms are reviewed. A Lagrangian approach to solve the general model is detailed, and, although the emphasis is on discrete models, some information on continuous covering is provided at the end of the chapter.
Drones can assist in mitigating traffic accidents by deterring reckless drivers, leveraging their flexible mobility. In the real-world, drones are fundamentally limited by their battery/fuel capacity and have to be replenished during long operations. In this article, we propose a novel approach where police cruisers act as mobile replenishment providers in addition to their traffic enforcement duties. We propose a binary integer linear program for determining the optimal rendezvous cruiser-drone enforcement policy, which guarantees that all drones are replenished on time and minimizes the likelihood of accidents. In an extensive empirical evaluation, we first show that human drivers are expected to react to traffic enforcement drones similar to how they react to police cruisers, using a first-of-its-kind human study in realistic simulated driving. Then, we show that our proposed approach significantly outperforms the common practice of constructing stationary replenishment installations using both synthetic and real-world road networks. Finally, we propose and evaluate a novel optimization speedup method for mitigating the increased runtime of our proposed approach.
In this paper we propose a dual ascent heuristic for solving the linear relaxation of the generalized set partitioning problem with convexity constraints, which often models the master problem of a column generation approach. The generalized set partitioning problem contains at the same time set covering, set packing and set partitioning constraints. The proposed dual ascent heuristic is based on a reformulation and it uses Lagrangian relaxation and subgradient method. It is inspired by the dual ascent procedure already proposed in literature, but it is able to deal with right hand side greater than one, together with under and over coverage. To prove its validity, it has been applied to the minimum sum coloring problem, the multi-activity tour scheduling problem, and some newly generated instances. The reported computational results show the effectiveness of the proposed method.
With shrinking feature sizes detecting small delay faults is getting more and more important. But not all small delay faults are detectable during at-speed test. By overclocking the circuit with several different test frequencies faster-than-at-speed test (FAST) is able to detect these hidden delay faults. If the clock frequency is increased, some outputs of the circuit may not have stabilized yet, and these outputs have to be considered as unknown ([Formula: see text]-values). These [Formula: see text]-values impede the test response compaction. In addition, the number and distribution of the [Formula: see text]-values vary with the clock frequency, and thus a very flexible [Formula: see text]-handling is needed for FAST. Most of the state-of-the-art solutions are not designed for these varying [Formula: see text]-profiles. Yet, the stochastic compactor by Mitra et al. can be adjusted to changing environments. It is easily programmable because it is controlled by weighted pseudo-random signals. But an optimal setup cannot be guaranteed in a FAST scenario. By partitioning the compactor into several smaller ones and a proper mapping of the scan outputs to the compactor inputs, the compactor can be better adapted to the varying [Formula: see text]-profiles. Finding the best setup can be formulated as a set partitioning problem. To solve this problem, several algorithms are presented. Experimental results show that independent from the scan chain configuration, the number of [Formula: see text]-values can be reduced significantly while the fault efficiency can be maintained. Additionally, it is shown that [Formula: see text]-reduction and fault efficiency can be adapted to user-defined goals.
: This memorandum presents a reformulation of the essentials of Balas' algorithm for the zero-one integer linear programming problem, and is based upon the idea of 'elementary tree search' that has also been used by Glover as the basis for his multiphase-dual algorithm. The present reformulation requires considerably less computer storage than the original version, and clarifies the rationale behind the algorithm, thereby leading naturally to variants and extensions.
In earlier papers [Opns. Res. 9, 849–859 (1961), and 11, 863–888 (1963)] the one-dimensional cutting stock problem was discussed as a linear programming problem. There it was shown how the difficulty of the enormous number of columns occurring in the linear programming formulation could be overcome by solving a knapsack problem at every pivot step. In this paper higher dimensional cutting stock problems are discussed as linear programming problems. The corresponding difficulty of the number of columns cannot in general be overcome for there is no efficient method for solving the generalized knapsack problem of the higher dimensional problem. However a wide class of cutting stock problems of industry have restrictions that permit their generalized knapsack problem to be efficiently solved. All of the cutting stock problems that yield to this treatment are ones in which the cutting is done in stages. In treating these practical cutting problems, one often encounters additional conditions that affect the solution. An example of this occurs in the cutting of corrugated boxes, which involves an auxiliary sequencing problem. This problem is discussed in some detail, and a solution described for the sequencing problem under given simplifying assumptions.
Polyhedral annexation is a new approach for generating all valid inequalities in mixed integer and combinatorial programming. These include the facets of the convex hull of feasible integer solutions. The approach is capable of exploiting the characteristics of the feasible solution space in regions both adjacent to and distant from the linear programming vertex without resorting to specialized notions of group theory, convex analysis or projective geometry. The approach also provides new ways for exploiting the branching inequalities of branch and bound.
We consider combinatorial programming problems of the form (IP): max{cx|Ax≤e, xj=0 or 1, ∀j, where A is a mxn matrix of zeroes and ones, e is a column vector of m ones and c is an arbitrary (non-negative) vector of n reals. This class of problems is known as the set packing problem, see e.g. (1). It is closely related to the set partitioning problem (SPP) and to the set covering problem. In the former case, the inequality constraints Ax≤e of (IP) are replaced by equality constraints Ax=e, whereas in the latter case one requires the constraints to hold with reversed inequality, i.e. Ax≥e. With respect to the set partitioning problem (SPP), it can be shown, that by appropriately modifying the objective function, the problem (SPP) can always be transformed into the form (IP) above. This is, however, not true in general if the set covering problem is considered. It is true, however, if the matrix A has at most two +1 entries per row, i.e. if the set covering problem assumes the special form of a node-covering problem in a (finite undirected) graph. As has been noted in (2), some of the structural properties of set partitioning and set packing problems do not carry over to the (general) set covering problem.
We generalize a method for constructing facets for the convex hull of integer solutions to set packing problems to arbitrary zero-one problems having nonnegative constraint-matrices. A particular class of facets is obtained explicitly and illustrated by a numerical example.
This paper gives an enumerative algorithm for the set-partitioning problem, that is, the set-covering problem with equality constraints, and presents computational results for real and randomly generated problems. The fact that many problems can be solved more rapidly than the corresponding linear programs demonstrates the efficiency of the algorithm; for example, a randomly generated problem with 1,400 variables and 100 constraints was solved in 15 minutes.
This paper establishes some useful properties of the equality-constrained set-covering problem P and the associated linear program P′. First, the Dantzig property of transportation matrices is shown to hold for a more general class of matrices arising in connection with adjacent integer solutions to P′. Next, we show that, for every feasible integer basis to P′, there are at least as many adjacent feasible integer bases as there are nonbasic columns. Finally, given any two basic feasible integer solutions x1 and x2 to P′, x2 can be obtained from x1 by a sequence of p pivots (where p is the number of indices j ϵ N for which xj1 is nonbasic and xj2 = 1), such that each solution in the associated sequence is feasible and integer. Some of our results have been conjectured earlier by Andrew, Hoffmann, and Krabek in a paper presented to ORSA in 1968.
A systematic procedure is presented for writing a Boolean function as a minimum sum of products. This procedure is a simplification and extension of the method presented by W. V. Quine. Specific attention is given to terms which can be included in the function solely for the designer's convenience.
This is a survey of the different approaches studied by a number of airlines in the past few years to attempt to optimize the allocation of crews to flights. The approach is usually one of integer programming with 0-1 variables, the matrix of coefficients having a very special form. The survey covers the generation, costing, reduction, and optimization of such matrices.
In an earlier paper [Opns. Res. 20 1153–1161 (1972)] we proved that any feasible integer solution to the linear program associated with the equality-constrained set-covering problem can be obtained from any other feasible integer solution by a sequence of less than m pivots (where m is the number of equations), such that each solution generated in the sequence is integer. However, degeneracy makes it difficult to find a sequence of pivots leading to an integer optimum. In this paper we give a constructive characterization of adjacency relations between integer vertices of the feasible set that enables us to generate edges (all, if necessary) connecting a given integer vertex to adjacent integer vertices. This helps overcome the difficulties caused by degeneracy and leads to a class of algorithms, of which we discuss two.
This paper presents an algorithm for the set-covering problem that is, min c'y: Ey ⧠e, y ⧠0, yi integer, where E is an m by n matrix of l's and 0's, and e is an m-vector of l's. The special problem structure permits a rather efficient, yet simple, solution procedure that is basically a 0, 1 search of the single-branch type coupled with linear programming and a suboptimization technique. The algorithm has been found to be highly effective for a good number of relatively large problems. Problems from 30 to 905 variables with as many as 200 rows have been solved in less than 16 minutes on an IBM 360 Model 50 computer. The algorithm's effectiveness stems from an efficient suboptimization procedure, which constructs excellent integer solutions from the solutions to linear-programming subproblems.
Previous papers by Fleischmann and by Lemke and Spielberg have reported computational experience with different algorithms for zero-one programming problems. This note supplements these papers by describing the experience with a special zero-one programming problem that arose from a mining context.
Given a set of jobs that must be performed, although not necessarily concurrently, a set of resources to perform the jobs, a set of cost measures for the resources, and a set of restrictions on the resources that may be used, the selection problem is to find a least-cost subset of resources that satisfies the restrictions and is capable of performing all the jobs. This selection problem is generalized into minimizing a linear function of Boolean variables subject to Boolean restriction equations. A "smallest" complete set of solutions is found that contains all optimal solutions.
An algorithm is proposed for solving linear programs with variables constrained to take only one of the values 0 or 1. It starts by setting all the n variables equal to 0, and consists of a systematic procedure of successively assigning to certain variables the value 1, in such a way that after trying a small part of all the 2n possible combinations, one obtains either an optimal solution, or evidence of the fact that no feasible solution exists. The only operations required under the algorithm are additions and subtractions; thus round-off errors are excluded. Problems involving up to 15 variables can be solved with this algorithm by hand in not more than 3-4 hours. An extension of the algorithm to integer linear programming and to nonlinear programming is available, but not dealt with in this article.
This paper views the location of emergency facilities as a set covering problem with equal costs in the objective. The sets are composed of the potential facility points within a specified time or distance of each demand point. One constraint is written for each demand point requiring “cover,” and linear programming is applied to solve the covering problem, a single-cut constraint being added as necessary to resolve fractional solutions.
The generalized lattice-point problem, posed by Charnes and studied by M. J. L. Kirby, H. Love, and others, is a linear program whose solutions are constrained to be extreme points of a specified polytope. We show how to exploit this and more general problems by convexity (or intersection) cut strategies without resorting to standard problem-augmenting techniques such as introducing 0-1 variables. In addition, we show how to circumvent “degeneracy” difficulties inherent in this problem without relying on perturbation (which provides uselessly shallow cuts) by identifying nondegenerate subregions relative to which cuts may be defined effectively. Finally, we give results that make it possible to obtain strengthening cuts for problems with special structures.
This study shows how the problem of optimally selecting files for extracting from an overlapping, multiple file data storage system can be treated as an integer programming problem and solved by means of existing integer programming computer routines. It presents several heuristic file selection problems and suggests how existing integer programming routines could be used in an extracting system. The study concludes that the integer programming formulation is a feasible means of expressing the file selection problem and that existing computer programs may afford useful methods of solution. The file selection problem gives an especially simple integer programming problem; available algorithms, modified to account for this simple structure may be expected to eliminate any chance of failure in operation.
This paper gives an elementary, inductive proof-“graphical” in spirit-of a theorem of Edmonds' which specifies the convex hull of the matchings of an arbitrary, finite, undirected graph in terms of a finite system of linear inequalities.
This paper studies the optimal utilization of stewardesses and optimal allocation of trips to bases by a special approximation to the solution of a linear programming problem.
In an earlier paper [Pierce, J. F. 1968. Application of combinatorial programming to a class of all-zero-one integer programming problems. Management Sci. 15 (3, November) 191–209.] combinatorial programming procedures were presented for solving a class of integer programming problems in which all elements are zero or one. By representing the problem elements in a binary computer as bits in a word and employing logical “and” and “or” operations in the problem-solving process, a number of problems involving several hundred integer variables were solved in a matter of seconds.
In the present paper a number of improvements in these earlier algorithms are presented, including a new search strategy, methods for reducing the original problem, and mechanisms for feasibility filtering in multi-word problems. With these improvements problem-solving efficiency has been increased in many instances by an order of magnitude. In addition, the present paper contains computational experience obtained in solving problems for the k-best solutions.
An algorithm is given which finds all optimal solutions, for a given set of criteria, to political redistricting problems. Using "population units" as indivisible elements, the first phase generates all feasible districts, where feasibility indicates contiguity, compactness and limited population deviation. The second phase finds that set of M feasible districts which "covers" each population unit exactly once, and minimizes the maximum deviation of any district population from the mean district population. Computational results indicate that states with 40 counties or fewer can be solved in less than 10 minutes on an IBM 7094. However, our attempt to solve a 55 county state was unsuccessful.
Recent models of location are drawn together and compared as to structure, criteria, and constraints. Private sector models are distinguished as those in which the total cost of transport and facilities is isolated as the objective to be minimized. The solution techniques of six such models are discussed. Public sector models are characterized by a criterion function involving a surrogate for social utility and by a constraint on investment in facilities or on the number of facilities. Five models with this format are discussed and compared. The two types of problem, location in the private sector and location in the public sector, are seen to have the same conceptual foundation, but formats which differ of necessity due to our inability to relate social utility to dollar value.
Problem-solving procedures based on the methods of combinatorial programming are presented for solving a class of integer programming problems in which all elements are zero or one. All of the procedures seek first a feasible solution and then successively better and better feasible solutions until ultimately one is discovered which is shown to be optimal. By representing the problem elements in a binary computer as bits in a word and employing logical "and" and "or" operations in the problem-solving process, a number of problems involving several hundred integer variables have been solved in a matter of seconds.
This paper presents the theoretical foundations for a new approach to integer programming. On the basis of the theoretical arguments, various specific computational procedures are developed which could be used to solve the integer programming problem. However, definitive computational results are not presented. In future papers we plan to explore the various computational options developed here.
This paper attempts to present the major methods, successful or interesting uses, and computational experience relating to integer or discrete programming problems. Included are descriptions of general algorithms for solving linear programs in integers, as well as some special purpose algorithms for use on highly structured problems. This reflects a belief, on the author's part, that various clever methods of enumeration and other specialized approaches are the most efficacious means existent by which to obtain solutions to practical problems. A serious try at gathering computational experience has been made—but facts are difficult to uncover.
The paper is written with intent to enable readers to read selected sections without having to read the whole.
The area of railroad scheduling and associated problems offers a potentially fruitful field for research in management science. The mass of data; the need for considering numerous aspects of the problem simultaneously; and the variety of restrictions (arising from the technology, labor agreements and government regulation) make the control of day to day operations a management problem of great complexity. The problems at this level are further compounded by the need for developing, at the same time, methods for planning and evaluating major changes in facilities. Such methods, if they are to be effective, must be capable of tracing out the implications of these changes on the day to day operation and appraising the results in the light of the long-run objectives of the railroad. This paper is an account of an attempt to apply certain techniques of management science to some of these scheduling problems as they were found to exist on a large terminal switching railroad.
In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form . This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.
The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. Some connections are established between prime implicants of a monotone or a regular Boolean function on the one hand, and facets of the convex hullH of the zeros of on the other. In particular (Corollary 2) a necessary and sufficient condition is given for a constraint of a covering problem to be a facet of the corresponding integer polyhedron. For any prime implicantP of, a nonempty familyF(P) of facets ofH is constructed. Proposition 17 gives easy-to-determine sharp upper bounds for the coefficients of these facets when is regular. A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet ofH, and this facet has 0–1 coefficients. Every nontrivial facet ofH with 0–1 coefficients is obtained from this class.
We consider two convex polyhedra related to the vertex packing problem for a finite, undirected, loopless graphG with no multiple edges. A characterization is given for the extreme points of the polyhedron\(\mathcal{L}_G = \{ x \in R^n :Ax \leqslant 1_m ,x \geqslant 0\} \), whereA is them × n edge-vertex incidence matrix ofG and 1m
is anm-vector of ones. A general class of facets of
= convex hull{x∈R
n
:Ax≤1
m
,x binary} is described which subsumes a class examined by Padberg [13]. Some of the results for
are extended to a more general class of integer polyhedra obtained from independence systems.
A type of program is considered in which, apart from the usual linear constraints, it is required that at least one variable from each of several sets be equal to zero. Applications include complementary pivot theory and concave minimization problems. Cutting planes are generated for the solution of such programs. A geometrical description of the cutting planes explains their meaning.
A necessary and sufficient condition is given for an inequality with coefficients 0 or 1 to define a facet of the knapsack polytope, i.e., of the convex hull of 0–1 points satisfying a given linear inequality. A sufficient condition is also established for a larger class of inequalities (with coefficients not restricted to 0 and 1) to define a facet for the same polytope, and a procedure is given for generating all facets in the above two classes. The procedure can be viewed as a way of generating cutting planes for 0–1 programs.
A zero–one matrix is called perfect if the polytope of the associated set packing problem has integral vertices only. By this definition, all totally unimodular zero–one matrices are perfect. In this paper we give a characterization of perfect zero–one matrices in terms offorbidden submatrices. Perfect zero–one matrices are closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge. Furthermore, the results obtained here bear on an unsolved problem in graph theory, the strong perfect graph conjecture, also due to C. Berge.
This paper studies some properties of hypergraphs in connection with a class of integer linear programming problems. The main result (theorem 3) states that the strong chromatic number of a balanced hypergraph is equal to its rank; this generalizes a result known for unimodular hypergraphs. Two applications of this result are given, the first one to Graph theory (theorem 5), the second one to integral linear programming (theorem 6).
Some of the main notions and theorems about blocking pairs of polyhedra and antiblocking pairs of polyhedra are described. The two geometric duality theories conform in many respects, but there are certain important differences. Applications to various combinatorial extremum problems are discussed, and some classes of blocking and anti-blocking pairs that have been explicitly determined are mentioned.
Given a linear inequality in 0–1 variables we attempt to obtain the faces of the integer hull of 0–1 feasible solutions. For the given inequality we specify how faces of a variety of lower-dimensional inequalities can be raised to give full-dimensional faces. In terms of a set, called a “strong cover”, we obtain necessary and sufficient conditions for any inequality with 0–1 coefficients to be a face, and characterize different forms that the integer hull must take.
In general the suggested procedures fail to produce the complete integer hull. Special subclasses of inequalities for which all faces can be generated are demonstrated. These include the “matroidal” and “graphic” inequalities, where a count on the number of such inequalities is obtained, and inequalities where all faces can be derived from lower dimensional faces.
A class of polytopes is defined which includes the polytopes related to the assignment problem, the edge-matching problem on complete graphs, the multi-dimensional assignment problem, and many other set partitioning problems. Modifying some results due to Balas and Padberg, we give a constructive proof that the diameter of these polytopes is less than or equal to two. This result generalizes a result obtained by Balinski and Rusakoff in connection with the assignment problem.
Furthermore, it is shown that the polytope associated with the travelling salesman problem has a diameter less than or equal to two. A weaker form of the Hirsch conjecture is also shown to be true for this polytope.