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Minimizing Latency in Online Ride and Delivery Services
Abstract
Motivated by the popularity of online ride and delivery services, we study natural variants of classical multi-vehicle minimum latency problems where the objective is to route a set of vehicles located at depots to serve request located on a metric space so as to minimize the total latency. In this paper, we consider point-to-point requests that come with source-destination pairs and release-time constraints that restrict when each request can be served. The point-to-point requests and release-time constraints model taxi rides and deliveries. For all the variants considered, we show constant-factor approximation algorithms based on a linear programming framework. To the best of our knowledge, these are the first set of results for the aforementioned variants of the minimum latency problems. Furthermore, we provide an empirical study of heuristics based on our theoretical algorithms on a real data set of taxi rides.
... Empty vehicle redistribution is explored by time windows for autonomous taxis and personal rapid transit from passengers' angles [4], to enhance methods by expected arrival/waiting times. Source-destination pairs are studied [5] as release-time limits in taxi rides/deliveries, for constantfactor approximation with linear programming to minimize latency. Optimal paths are found for taxi carpooling by genetic algorithms [6]. ...
Pillars of smart cities include smart environment, mobility and economy. We explore impacts on these to enhance smart cities, heading towards a smart planet. Our motivation emerges from the need to decarbonize transportation. In this context, ride-sharing companies deploy electric vehicles (EVs). These should be managed by various factors: battery demand, EV charging station location, service availability, and charging time. Ride-sharing EVs aim to maximize profits via more rides. Our paper explores game theory in AI here. We propose E-Ride-Minimax, adapting the Minimax algorithm, treating EV ride-sharing companies as players. We hypothesize one player choosing its next move via total passenger-travel distance (longer the distance, larger the profit); and another player via battery usage (ratio of total passenger-travel distance to vehicle-passenger distance: optimizing this ratio enables more travel without recharging). Experimental results reveal that rising passenger numbers yield maximum battery savings (e.g. rush hours / major events); followed by stable and falling numbers. Our findings indicate that E-Ride-Minimax can reduce battery usage in some circumstances by 64%, losing 1% profit. This is vital, given global emphasis on climate change. It increases cost-effectiveness, consumer participation and passengers per mile; reduces energy use and greenhouse gas emissions; and thus helps decarbonize transportation.
... Some examples of these works include: Cordeau and Laporte [16], who performed a review of scientific literature on the dial-a-ride problem, which consists on scheduling the pickup and delivery requests for a group of vehicles to transport a set of users, taking into account the users origins and destinations. Similarly, Das et al. [17] dealt with the the multi-vehicle minimum latency problem by proposing an approximation algorithm with point-to-point requests, which enabled the minimizing of the total latency while serving all requests. Lastly, in [18], the authors contributed to solving the multi-agent pickup and delivery problem on a warehouse scenario by proposing two heuristic algorithms capable of simultaneously performing the task-allocation and path-planning for mobile robots. ...
Rideshare platforms such as Uber and Lyft dynamically dispatch drivers to match riders’ requests. We model the dispatching process in rideshare as a Markov chain that takes into account the geographic mobility of both drivers and riders over time. Prior work explores dispatch policies in the limit of such Markov chains; we characterize when this limit assumption is valid, under a variety of natural dispatch policies. We give explicit bounds on convergence in general, and exact (including constants) convergence rates for special cases. Then, on simulated and real transit data, we show that our bounds characterize convergence rates—even when the necessary theoretical assumptions are relaxed. Additionally these policies compare well against a standard reinforcement learning algorithm which optimizes for profit without any convergence properties.
Motivated by the popularity of online ride and delivery services, we study natural variants of classical multi-vehicle minimum latency problems where the objective is to route a set of vehicles located at depots to serve requests located on a metric space so as to minimize the total latency. In this paper, we consider point-to-point requests that come with source-destination pairs and release-time constraints that restrict when each request can be served. The point-to-point requests and release-time constraints model taxi rides and deliveries. For all the variants considered, we show constant-factor approximation algorithms based on a linear programming framework. To the best of our knowledge, these are the first set of results for the aforementioned variants of the minimum latency problems. Furthermore, we provide an empirical study of heuristics based on our theoretical algorithms on a real data set of taxi rides.
In this paper we propose five mathematical formulations for the multiple minimum latency problem. The first three mathematical models are straight derived from classical formulations and from a flow-based formulation to the multiple travelling salesman problem. The last two are obtained as generalizations of time-dependent formulations to the minimum latency problem. We carry out an extensive computational experimentation to evaluate the performance of the proposed models using routing and scheduling instances. These experiments evidence that the time-dependent formulations show a much better performance than the other formulations, regarding to the size of instances that can be solved and the elapsed computational time to reach the optimal solutions. The obtained results suggest to consider the development of time-dependent formulations for other problems that consider the latency as objective function.
Recommending routes for a group of competing taxi drivers is almost untouched in most route recommender systems. For this kind of problem, recommendation fairness and driving efficiency are two fundamental aspects. In the paper, we propose SCRAM, a sharing considered route assignment mechanism for fair taxi route recommendations. SCRAM aims to provide recommendation fairness for a group of competing taxi drivers, without sacrificing driving efficiency. By designing a concise route assignment mechanism, SCRAM achieves better recommendation fairness for competing taxis. By considering the sharing of road sections to avoid unnecessary competition, SCRAM is more efficient in terms of driving cost per customer (DCC). We test SCRAM based on a large number of historical taxi trajectories and validate the recommendation fairness and driving efficiency of SCRAM with extensive evaluations. Experimental results show that SCRAM achieves better recommendation fairness and higher driving efficiency than three compared approaches.
In this paper we study a dynamic problem of ridesharing and taxi sharing with time windows. We consider a scenario where people needing a taxi or interested in getting a ride use a phone app to designate their source and destination points in a city, as well others restrictions (such as maximum allowable time to be at the destination). On the other hand, we have taxis and people interested in giving a ride, with their current positions and also some constraints (vehicle capacity, destination, maximum time to destination). We want to maximize the number of shared trips: in the case of taxis, people going to close locations can share the costs of the trip, and in case of rides, the driver and passengers can share costs as well. This problem is dynamic since new calls for taxis or calls for rides arrive on demand. This gives rise to an optimization problem which we prove to be NP-Hard. We then propose heuristics to deal with it. We focus on the taxi sharing problem, but we show that our model is easily extendable to model the ridesharing situation or even a situation where there are both taxis and car owners. In addition, we present a framework that consists basically of a client application and a server. The last one processes all incoming information in order to match vehicles to passengers requests. The entire system can be used by taxi companies and riders in a way to reduce the traffic in the cities and to reduce the emission of greenhouse gases.
A travelling deliveryman needs to find a tour such that the total waiting time of all the customers he has to visit is minimum. The deliveryman starts his tour at a depot, travelling at constant velocity. In this paper we suggest a general variable neighborhood search based heuristic to solve this NP-hard combinatorial optimization problem. We combine several classical neighborhood structures and design data structure to store and update the incumbent solution efficiently. In this way, we are able to explore neighborhoods as efficiently as when solving the travelling salesman problem. Computational results obtained on usual test instances show that our approach outperforms recent heuristics from the literature.
Consider a complete directed graph in which each arc has a given length. There is a set of jobs, each job i located at some node of the graph, with an associated processing time hi, and whose execution has to start within a presepecified time window [ri, di]. We have a single server that can move on the arcs of the graph, at unit speed, and that has to execute all of the jobs within their respective time windows. We consider the following two problems: (a) minimize the time by which all jobs are executed (traveling salesman problem) and (b) minimize the sum of the waiting times of the jobs (traveling repairman problem). We focus on the following two special cases: (a) The jobs are located on a line and (b) the number of nodes of the graph is bounded by some integer constant B. Furthermore, we consider in detail the special cases where (a) all of the processing times are 0, (b) all of the release times ri are 0, and (c) all of the deadlines di are infinite. For many of the resulting problem combinations, we settle their complexity either by establishing NP-completeness or by presenting polynomial (or pseudopolynomial) time algorithms. Finally, we derive algorithms for the case where, for any time t, the number of jobs that can be executed at that time is bounded.
The Traveling Salesman Problem (TSP) is a classical prob- lem in discrete optimization. Its paradigmatic character makes it
one of the most studied in computer science and operations research and one for which an impressive amount of algorithms (in
particular heuristics and approximation algorithms) have been proposed. While in the general case the problem is known not
to allow any constant ratio approximation algorithm and in the metric case no better algorithm than Christofides’ algorithm
is known, which guarantees an approximation ratio of 3/2, re- cently an important breakthrough by Arora has led to the definition
of a new polynomial approximation scheme for the Euclidean case. A grow- ing attention has also recently been posed on the
approximation of other paradigmatic routing problems such as the Travelling Repairman Prob- lem (TRP). The altruistic Travelling
Repairman seeks to minimimize the average time incurred by the customers to be served rather than to mini- mize its working
time like the egoistic Travelling Salesman does. The new approximation scheme for the Travelling Salesman is also at the basis
of a new approximation scheme for the Travelling Repairman problem in the euclidean space. New interesting constant approximation
algorithms have recently been presented also for the Travelling Repairman on gen- eral metric spaces. Interesting applications
of this line of research can be found in the problem of routing agents over the web. In fact the prob- lem of programming
a “spider” for efficiently searching and reporting information is a clear example of potential applications of algorithms
for the above mentioned problems. These problems are very close in spirit to the problem of searching an object in a known
graph introduced by Koutsoupias, Papadimitriou and Yannakakis [14]. In this paper, moti- vated by web searching applications, we summarize the most important recent results concerning the
approximate solution of the TRP and the TSP and their application and extension to web searching problems.
In dial-a-ride problems, items have to be transported from a source to a destination. The characteristics of the servers involved as well as the specific requirements of the rides may vary. Problems are defined on some metric space, and the goal is to find a feasible solution that minimizes a certain objective function. The structure of these problems allows for a notation similar to the standard notation for scheduling and queueing problems. We introduce such a notation and show how a class of 7,930 dial-a-ride problem types arises from this approach. In examining their computational complexity, we define a partial ordering on the problem class and incorporate it in the computer program DARCLASS. As input DARCLASS uses lists of problems whose complexity is known. The output is a classification of all problems into one of three complexity classes: solvable in polynomial time, NP-hard, or open. For a selection of the problems that form the input for DARCLASS, we exhibit a proof of polynomial-time solvability or NP-hardness.
For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete. In contrast with these results, a linear time approximation algorithm for the clustering problem is presented.
Motivated by the popularity of online ride and delivery services, we study natural variants of classical multi-vehicle minimum latency problems where the objective is to route a set of vehicles located at depots to serve requests located on a metric space so as to minimize the total latency. In this paper, we consider point-to-point requests that come with source-destination pairs and release-time constraints that restrict when each request can be served. The point-to-point requests and release-time constraints model taxi rides and deliveries. For all the variants considered, we show constant-factor approximation algorithms based on a linear programming framework. To the best of our knowledge, these are the first set of results for the aforementioned variants of the minimum latency problems. Furthermore, we provide an empirical study of heuristics based on our theoretical algorithms on a real data set of taxi rides.
In this paper, we study a k-Travelling Repairmen Problem where the objective is to minimize the sum of clients waiting time to receive service. This problem is relevant in applications that involve distribution of humanitarian aid in disaster areas, delivery and collection of perishable products and personnel transportation, where reaching demand points to perform service, fast and fair, is a priority. This paper presents a new mixed integer formulation and a simple and efficient metaheuristic algorithm. The proposed formulation consumes less computational time and allows solving to optimality more than three times larger data instances than the previous formulation published in literature, even outperforming a recently published Branch and Price and Cut algorithm for this problem. The proposed metaheuristic algorithm solved to optimality 386 out of 389 tested instances in a very short computational time. For larger instances, the algorithm was assessed using the best results reported in the literature for the Cumulative Capacitated Vehicle Routing Problem.
We consider various {\em multi-vehicle versions of the minimum latency
problem}. There is a fleet of $k$ vehicles located at one or more depot nodes,
and we seek a collection of routes for these vehicles that visit all nodes so
as to minimize the total latency incurred, which is the sum of the client
waiting times. We obtain an $8.497$-approximation for the version where
vehicles may be located at multiple depots and a $7.183$-approximation for the
version where all vehicles are located at the same depot, both of which are the
first improvements on this problem in a decade. Perhaps more significantly, our
algorithms exploit various LP-relaxations for minimum-latency problems. We show
how to effectively leverage two classes of LPs---{\em configuration LPs} and
{\em bidirected LP-relaxations}---that are often believed to be quite powerful
but have only sporadically been effectively leveraged for network-design and
vehicle-routing problems. This gives the first concrete evidence of the
effectiveness of LP-relaxations for this class of problems. The
$8.497$-approximation the multiple-depot version is obtained by rounding a
near-optimal solution to an underlying configuration LP for the problem. The
$7.183$-approximation can be obtained both via rounding a bidirected LP for the
single-depot problem or via more combinatorial means. The latter approach uses
a bidirected LP to obtain the following key result that is of independent
interest: for any $k$, we can efficiently compute a rooted tree that is at
least as good, with respect to the prize-collecting objective (i.e., edge cost
+ number of uncovered nodes) as the best collection of $k$ rooted paths. Our
algorithms are versatile and extend easily to handle various extensions
involving: (i) weighted sum of latencies, (ii) constraints specifying which
depots may serve which nodes, (iii) node service times.
The GPS technology and new forms of urban geography have changed the paradigm for mobile services. As such, the abundant availability of GPS traces has enabled new ways of doing taxi business. Indeed, recent efforts have been made on developing mobile recommender systems for taxi drivers using Taxi GPS traces. These systems can recommend a sequence of pick-up points for the purpose of maximizing the probability of identifying a customer with the shortest driving distance. However, in the real world, the income of taxi drivers is strongly correlated with the effective driving hours. In other words, it is more critical for taxi drivers to know the actual driving routes to minimize the driving time before finding a customer. To this end, in this paper, we propose to develop a cost-effective recommender system for taxi drivers. The design goal is to maximize their profits when following the recommended routes for finding passengers. Specifically, we first design a net profit objective function for evaluating the potential profits of the driving routes. Then, we develop a graph representation of road networks by mining the historical taxi GPS traces and provide a Brute-Force strategy to generate optimal driving route for recommendation. However, a critical challenge along this line is the high computational cost of the graph based approach. Therefore, we develop a novel recursion strategy based on the special form of the net profit function for searching optimal candidate routes efficiently. Particularly, instead of recommending a sequence of pick-up points and letting the driver decide how to get to those points, our recommender system is capable of providing an entire driving route, and the drivers are able to find a customer for the largest potential profit by following the recommendations. This makes our recommender system more practical and profitable than other existing recommender systems. Finally, we carry out extensive experiments on a real-world data set collected from the San Francisco Bay area and the experimental results clearly validate the effectiveness of the proposed recommender system.
In this paper, we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel; the resulting problem is called the multiple traveling repairmen problem with distance constraints (MTRPD). In the MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers. Each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once. The objective is to minimize the total waiting time of all customers after the vehicles leave the depot. To optimally solve the MTRPD, we propose a new exact branch-and-price-and-cut algorithm, where the column generation pricing subproblem is a resource-constrained elementary shortest-path problem with cumulative costs. An ad hoc label-setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem. Computational results show the effectiveness of the proposed method. The optimal solutions to 179 out of 180 test instances are reported in this paper. Our computational results serve as benchmarks for future researchers on the problem.
People often have the demand to decide where to wait for a taxi in order to save their time. In this paper, to address this problem, we employ the non-homogeneous Poisson process (NHPP) to model the behavior of vacant taxis. According to the statistics of the parking time of vacant taxis on the roads and the number of the vacant taxis leaving the roads in history, we can estimate the waiting time at different times on road segments. We also propose an approach to make recommendations for potential passengers on where to wait for a taxi based on our estimated waiting time. Then we evaluate our approach through the experiments on simulated passengers and actual trajectories of 12,000 taxis in Beijing. The results show that our estimation is relatively accurate and could be regarded as a reliable upper bound of the waiting time in probability. And our recommendation is a tradeoff between the waiting time and walking distance, which would bring practical assistance to potential passengers. In addition, we develop a mobile application TaxiWaiter on Android OS to help the users wait for taxis based on our approach and historical data.
We give a polynomial time, $(1+\epsilon)$-approximation algorithm for the
traveling repairman problem (TRP) in the Euclidean plane, on weighted planar
graphs, and on weighted trees. This improves on the known quasi-polynomial time
approximation schemes for these problems. The algorithm is based on a simple
technique that reduces the TRP to what we call the \emph{segmented TSP}. Here,
we are given numbers $l_1,\dots,l_K$ and $n_1,\dots,n_K$ and we need to find a
path that visits at least $n_h$ points within path distance $l_h$ from the
starting point for all $h\in\{1,\dots,K\}$. A solution is $\alpha$-approximate
if at least $n_h$ points are visited within distance $\alpha l_h$. It is shown
that any algorithm that is $\alpha$-approximate for \emph{every constant} K in
some metric space, gives an $\alpha(1+\epsilon)$-approximation for the TRP in
the same metric space. Subsequently, approximation schemes are given for this
segmented TSP problem in different metric spaces.The segmented TSP with only
one segment (K=1) is equivalent to the k-TSP for which a
$(2+\epsilon)$-approximation is known for a general metric space. Hence, this
approach through the segmented TSP gives new impulse for improving on the
3.59-approximation for TRP in a general metric space. A similar reduction
applies to many other minimum latency problems. To illustrate the strength of
this approach we apply it to the well-studied scheduling problem of minimizing
total weighted completion time under precedence constraints, $1|prec|\sum
w_{j}C_{j}$, and present a polynomial time approximation scheme for the case of
interval order precedence constraints. This improves on the known
3/2-approximation for this problem. Both approximation schemes apply as well if
release dates are added to the problem.
Many approximation results for single machine scheduling problems rely on the conversion of preemptive schedules into (preemptive
or non-preemptive) solutions. The initial preemptive schedule is usually an optimal solution to a combinatorial relaxation
or a linear programming relaxation of the scheduling problem under consideration. It therefore provides a lower bound on the
optimal objective function value. However, it also contains structural information which is useful for the construction of
provably good feasible schedules. In this context, list scheduling in order of so-called α-points has evolved as an important and successful tool. We give a survey and a uniform presentation of several approximation
results for single machine scheduling with total weighted completion time objective from the last years which rely on the
concept of α-points.
We present a polynomial-time approximation algorithm with worst-case ratio 7/6 for the special case of the traveling salesman problem in which all distances are either one or two. We also show that this special case of the traveling salesman problem is MAX SNP-hard, and therefore it is unlikely that it has a polynomial-time approximation scheme.
The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.
The Minimum Latency Problem (MLP) is a variant of the Traveling Salesman Problem which aims to minimize the sum of arrival times at vertices. The problem arises in a number of practical applications such as logistics for relief supply, scheduling and data retrieval in computer networks. This paper introduces a simple metaheuristic for the MLP, based on a greedy randomized approach for solution construction and iterated variable neighborhood descent with random neighborhood ordering for solution improvement. Extensive computational experiments on nine sets of benchmark instances involving up to 1000 customers demonstrate the good performance of the method, which yields solutions of higher quality in less computational time when compared to the current best approaches from the literature. Optimal solutions, known for problems with up to 50 customers, are also systematically obtained in a fraction of seconds.
The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.
We study a variant of the maximum coverage problem which we label the maximum coverage problem with group budget constraints (MCG). We are given a collection of sets \({\cal S} = \{S_1, S_2, \ldots, S_m\}\) where each set S
i
is a subset of a given ground set X. In the maximum coverage problem the goal is to pick k sets from \({\cal S}\) to maximize the cardinality of their union. In the MCG problem \({\cal S}\) is partitioned into groupsG
1, G
2, ..., G
ℓ. The goal is to pick k sets from \({\cal S}\) to maximize the cardinality of their union but with the additional restriction that at most one set be picked from each group. We motivate the study of MCG by pointing out a variety of applications. We show that the greedy algorithm gives a 2-approximation algorithm for this problem which is tight in the oracle model. We also obtain a constant factor approximation algorithm for the cost version of the problem. We then use MCG to obtain the first constant factor approximation algorithms for the following problems: (i) multiple depot k-traveling repairmen problem with covering constraints and (ii) orienteering problem with time windows when the number of time windows is a constant.
In the minimum latency problem (MLP) we are given n points v
1,..., v
n and a distance d(v
i,v
j) between any pair of points. We have to find a tour, starting at v
1 and visiting all points, for which the sum of arrival times is minimal. The arrival time at a point v
i is the traveled distance from v
1 to v
i in the tour. The minimum latency problem is MAX-SNP-hard for general metric spaces, but the complexity for the problem where the metric is given by an edge-weighted tree has been a long-standing open problem. We show that the minimum latency problem is NP-hard for trees even with weights in {0,1}.
We study three combinatorial optimization problems related to searching a graph that is known in advance, for an item that resides at an unknown node. The search ratio of a graph is the optimum competitive ratio (the worst-case ratio of the distance traveled before the unknown node is visited, over the distance between the node and a fixed root, minimized over all Hamiltonian walks of the graph). We also define the randomized search ratio (we minimize over all distributions of permutations). Finally, the traveling repairman problem seeks to minimize the expected time of visit to the unknown node, given some distribution on the nodes. All three of these novel graph-theoretic parameters are NP-complete —and MAXSNP-hard — to compute exactly; we present interesting approximation algorithms for each. We also show that the randomized search ratio and the traveling repairman problem are related via duality and polyhedral separation.
The minimum latency problem(MLP) is a well-studied variant of the traveling salesman problem (TSP). In the MLP, the server's goal is to minimize the average latency that the clients experience prior to being served, rather than the total latency experienced by the server (as in the TSP). The MLP sometimes goes by other names, such as the traveling repairman problem, or thedeliveryman problem. Unlike most combinatorial optimization problems, the MLP is NP-hard even on trees (Sitters, 2001). Our main result is an improved approximation algorithm for the MLP on trees, upon which we build improved approximation algorithms for a much wider class of graphs. The MLP on trees is interesting for several reasons. First, many of the aspects that make the problem difficult on general graphs are already present in the tree case. Second, all existing approximation algorithms for general graphs are built on approxi- mation algorithms for the tree case. Third, there has been no im- provement for the tree case since the 3.59-approximation of Goe- mans and Kleinberg, first introduced 14 years ago in 1996. Fourth, in the intervening period, the best ratio for general metrics has been improved to match the 3.59 for trees (Chaudhuri et al., 2003). In this paper, we improve the approximation ratio for trees to 3.03. In fact, our 3.03-approximation algorithm works for any class of graphs in which the related prize-collecting stroll (PCS) problem is solvable in polynomial time, such as graphs of constant treewidth. More generally, for any class of graphs that admit a Lagrangian-preserving -approximation algorithm, we can use this algorithm as a black box to achieve a 3.03 -approximation for the MLP. Sadly, this does not immediately improve the ratio of 3.59 for general graphs, because the current best value of for that case is 2. One interesting piece of our analysis is the solution of an infinite-dimensional linear program, used to analyze a finite- dimensional factor-revealing linear program (FRLP). We believe that our methods may hold promise for easing the analysis of other FRLPs encountered in the literature.
In this article, we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to point-orienteering problem is the following: Given an edge-weighted graph G=(V, E) (directed or undirected), two nodes s, t ∈ V and a time limit B, find an s-t walk in G of total length at most B that maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such as k-MST. Orienteering with time-windows is the more general problem in which each node v has a specified time-window [R(v), D(v)] and a node v is counted as visited by the walk only if v is visited during its time-window. We design new and improved algorithms for the orienteering problem and orienteering with time-windows. Our main results are the following:
— A (2+ε) approximation for orienteering in undirected graphs, improving upon the 3-approximation of Bansal et al. [2004].
— An O(log² OPT) approximation for orienteering in directed graphs, where OPT ≤ n is the number of vertices visited by an optimal solution. Previously, only a quasipolynomial-time algorithm due to Chekuri and Pál [2005] achieved a polylogarithmic approximation (a ratio of O(log OPT)).
— Given an α approximation for orienteering, we show an O(α ċ max{log OPT, log lmax/lmin}) approximation for orienteering with time-windows, where lmax and lmin are the lengths of the longest and shortest time-windows respectively.
We consider the k-traveling repairman problem, a generalization of the metric traveling repairman problem, also known as the minimum latency problem, to multiple repairmen. We give an 8:497-approximation algorithm for this generalization, where denotes the best achievable approximation factor for the problem of nding the least cost rooted tree spanning i vertices (i-MST) problem. This can be compared with the best known approximation algorithm for the case k = 1, which is 3:59. We are aware of no previous work on the approximability of the present problem.
Abstract We give a 7.18-approximation algorithm for the minimum latency problem that uses only O(n log n) calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson.This improves the previous best algorithms in both performance guarantee and running time. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires anapproximation algorithm for the k-MST problem which is called as a black box for each valueof k. Their algorithm can achieve a performance guarantee of 10.77 while making O(n2 log n)PCST calls (via a k-MST algorithm of Garg), or a performance guarantee of 7.18 + ffl while using nO(1/ffl) PCST calls (via a k-MST algorithm of Arora and Karakostas). In all cases, the runningtime is dominated by the PCST calls. Since the PCST subroutine can be implemented to run in O(n2) time, the overall running time of our algorithm is O(n3 log n).The basic idea for our improvement is that we do not treat the k-MST algorithm as ablack box. This allows us to take advantage of some special situations in which the PCST
Given a tour visiting n points in a metric space, the latency of one of these points p is the distance traveled in the tour before reaching p. The minimum latency problem asks for a tour passing through n given points for which the total latency of the n points is minimum; in effect, we are seeking the tour with minimum average "arrival time." This problem has been studied in the operations research literature, where it has also been termed the "delivery-man problem" and the "traveling repairman problem." The approximability of the minimum latency problem was first considered by Sahni and Gonzalez in 1976; however, unlike the classical traveling salesman problem, it is not easy to give any constant-factor approximation algorithm for the minimum latency problem. Recently, Blum, Chalasani, Coppersmith, Pulleyblank, Raghavan, and Sudan gave the first such algorithm, obtaining an approximation ratio of 144. In this work, we present an algorithm which improves this ratio to 21:55. The dev...
We are given a set of points $p_1,\ldots , p_n$ and a symmetric distance matrix $(d_{ij})$ giving the distance between $p_i$ and $p_j$. We wish to construct a tour that minimizes $\sum_{i=1}^n \ell(i)$, where $\ell(i)$ is the {\em latency} of $p_i$, defined to be the distance traveled before first visiting $p_i$. This problem is also known in the literature as the {\em deliveryman problem} or the {\em traveling repairman problem}. It arises in a number of applications including disk-head scheduling, and turns out to be surprisingly different from the traveling salesman problem in character. We give exact and approximate solutions to a number of cases, including a constant-factor approximation algorithm whenever the distance matrix satisfies the triangle inequality. Comment: 9 pages
The time dependent traveling salesman problem: polyhedra and algorithm
- Hernán Abeledo
- Ricardo Fukasawa
- Artur Pessoa
- Eduardo Uchoa
Hernán Abeledo, Ricardo Fukasawa, Artur Pessoa, and Eduardo Uchoa. 2013.
The time dependent traveling salesman problem: polyhedra and algorithm.
Mathematical Programming Computation 5, 1 (01 Mar 2013), 27-55. https:
//doi.org/10.1007/s12532-012-0047-y
Multiagent Selforganization for a Taxi Dispatch System
- Aamena Alshamsi
- Sherief Abdallah
- Iyad Rahwan
Aamena Alshamsi, Sherief Abdallah, and Iyad Rahwan. 2009. Multiagent Selforganization for a Taxi Dispatch System. In Proceedings of the 8th International
Conference on Autonomous Agents and Multiagent Systems (AAMAS '09).
Approximation Schemes for Minimum Latency Problems
- Sanjeev Arora
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