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a The colored graph G. b A rainbow spanning forest composed by three rainbow trees. c A rainbow spanning forest composed by only two rainbow trees. This is the optimal solution for the RSFP on G
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Given an undirected and edge-colored graph G, a rainbow component of G is a subgraph of G having all the edges with different colors. The Rainbow Spanning Forest Problem consists of finding a spanning forest of G with the minimum number of rainbow components. The problem is known to be NP-hard on general graphs and on trees. In this paper, we prese...
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Citations
... Carrabs et al. [2] proposed an integer linear mathematical formulation for the RSF problem which is as follows: subject to (1) where c is a set of binary variables corresponding to each rainbow component (rainbow tree) c of a rainbow spanning forest, i.e., for c = 1, … , rt . If c = 1 iff c contains at least one vertex. ...
... If y v c = 1 iff the vertex v is a part of c. x e c is a set of binary variables. If x e c = 1 iff the edge e is a part of c. Carrabs et al. [2] mentioned that if no upper bound is known, then the value rt = |V|-1 can be set as an upper bound for the maximum number of connected components. (v) refers to the set of edges incident to v in G and E k = {e ∈ E ∶ l(e) = k} . ...
... The objective function for the RSF problem aims to minimize the number of rainbow trees mentioned in Eq. (1). Constraints (2) state that the number of vertices which belongs to a rainbow tree cannot be greater than the number of colors plus one. Constraints (3) ensure that a rainbow tree must have at least one vertex. ...
Given a connected, undirected and edge-colored graph, the rainbow spanning forest (RSF) problem aims to find a rainbow spanning forest with the minimum number of rainbow trees, where a rainbow tree is a connected acyclic subgraph of the graph whose each edge is associated with a different color. This problem is NP-hard and finds several applications in distinguishing among various types of connections. Being a grouping problem, this paper proposes a steady-state grouping genetic algorithm (SSGGA) for the RSF problem. To the best of our knowledge, this is the first work on steady-state grouping genetic algorithm for this problem. While keeping in view of grouping aspects of the problem, each individual, in the proposed SSGGA, is encoded as a group of rainbow trees, and accordingly, a problem-specific crossover operator is designed. Moreover, SSGGA uses the idea of two steps in its replacement scheme. All such elements of SSGGA coordinate effectively and overall help in finding high-quality solutions. Computational results obtained over a set of benchmark instances show that overall SSGGA, in terms of solution quality, is superior to all other existing approaches in the literature for this problem.
... Note that the Dijkstra algorithm has often been combined in solution approaches for solving some variants of the SPP, as, for example, in [23][24][25]. The second heuristic algorithm is an iterative neighborhood search procedure [26] based on an initial randomly generated solution, improved thanks to IPa previously cited. Among the hundred generated starting solutions, the best one obtained is given for comparison with the previous method. ...
This paper presents a new heuristic algorithm tailored to solve large instances of an NP-hard variant of the shortest path problem, denoted the cost-balanced path problem, recently proposed in the literature. The problem consists in finding the origin–destination path in a direct graph, having both negative and positive weights associated with the arcs, such that the total sum of the weights of the selected arcs is as close to zero as possible. At least to the authors’ knowledge, there are no solution algorithms for facing this problem. The proposed algorithm integrates a constructive procedure and an improvement procedure, and it is validated thanks to the implementation of an iterated neighborhood search procedure. The reported numerical experimentation shows that the proposed algorithm is computationally very efficient. In particular, the proposed algorithm is most suitable in the case of large instances where it is possible to prove the existence of a perfectly balanced path and thus the optimality of the solution by finding a good percentage of optimal solutions in negligible computational time.
... Many variants of SPP are known to be NP-hard; thus, in this section, before presenting a mathematical formulation for modeling and solving CBPP, the problem complexity is investigated. In particular, thanks to a reduction algorithm [21,22], it has been proved that the problem is NP-hard in its general form. ...
This paper introduces a new variant of the Shortest Path Problem (SPP) called the Cost-Balanced Path Problem (CBPP). Various real problems can either be modeled as BCPP or include BCPP as a sub-problem. We prove several properties related to the complexity of the CBPP problem. In particular, we demonstrate that the problem is NP-hard in its general version, but it becomes solvable in polynomial time in a specific family of instances. Moreover, a mathematical formulation of the CBPP, as a mixed-integer programming model, is proposed, and some additional constraints for modeling real requirements are given. This paper validates the proposed model and its extensions with experimental tests based on random instances. The analysis of the results of the computational experiments shows that the proposed model and its extension can be used to model many real applications. Obviously, due to the problem complexity, the main limitation of the proposed approach is related to the size of the instances. A heuristic solution approach should be required for larger-sized and more complex instances.
... Eiben et al. (2018) proved that the Strong Rainbow Vertex Coloring is NP-complete even on graphs of diameter 3. The problem consists in finding a node coloring for which given two nodes and , rainbow -path exists. The Rainbow Spanning Forest (Carrabs et al., 2017b) consists in finding a spanning forest such that all components (trees) are rainbow and the number of trees is minimized. Carrabs et al. (2017a) proved that the problem is NP-complete even on trees, but it is solvable in polynomial time on paths, cycles and if the optimal solution value is equal to 1. Moreno et al. (2019) proposed an improved integer linear programming model and a GRASP metaheuristic to solve it. ...
Given an undirected and edge-colored graph with non-negative edge lengths, the aim of the Rainbow Steiner Tree Problem (RSTP) is to find a minimum Steiner Tree that uses at most one edge for each color. In this paper, the RSTP is introduced, a mathematical model is proposed to formally represent the problem and its theoretical properties are investigated. Since the RSTP belongs to the NP-class, two heuristic methods are designed: a Lagrangian relaxation approach and a multistart algorithm.
Extensive computational experiments are carried out on a significant set of test problems to empirically evaluate the performance of the proposed approaches. The computational results show that the two approaches are both effective and efficient compared to the ILOG CPLEX solver.
... Matsypura, Veremyev, Prokopyev, and Pasiliao (2019), Bökler, Chimani, Wagner, and Wiedera (2020), and Marzo and Ribeiro (2021) considered the problem of encountering the longest induced path. Besides, integer programming approaches have been successfully applied to several optimization problems related to encountering trees and forests with certain properties (Melo, Samer, & Urrutia, 2016;Carrabs, Cerrone, Cerulli, & Silvestri, 2018;Carrabs, Cerulli, Pentangelo, & Raiconi, 2021). ...
Given a graph $G=(V,E)$ with a weight $w_v$ associated with each vertex $v\in V$, the maximum weighted induced forest problem consists of encountering a maximum weighted subset $V'\subseteq V$ of the vertices such that $V'$ induces a forest. This NP-hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domains. The closely related maximum weighted induced tree problem, on the other hand, requires that the subset $V'\subseteq V$ induces a tree. We propose two new integer programming formulations with an exponential number of constraints and branch-and-cut procedures for the maximum weighted induced forest problem. Furthermore, we show how formulations for the problem can be very easily adapted to obtain maximum weighted induced trees. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when implemented in a standard commercial MIP (mixed integer programming) solver. More specifically, five formulations are compared, two compact (i.e., with a polynomial number of variables and constraints) ones and the three others with an exponential number of constraints. The results indicate that one of the newly proposed formulations, denoted tree with cycle elimination (TCYC), outperforms those available in the literature when it comes to the average times for proving optimality for the small instances, especially the more challenging ones. Additionally, this formulation can achieve much lower average times for solving the larger random instances that can be optimally solved. The results also illustrate the impact of offering high quality initial feasible solutions in the performance of the formulations.
... Recently, the authors in Carrabs et al. (2018a) proved that this problem is NP-complete on trees and is solvable in polynomial time on paths, cycles and if the optimal solution value is equal to 1. Moreover, in Carrabs et al. (2018b), an integer mathematical formulation and a multi-start scheme based on a greedy algorithm were presented for this problem. They studied multicolored graphs with different densities and colors. ...
... In this work, we present a modification of the mathematical formulation presented in Carrabs et al. (2018b), as well as a fast GRASP metaheuristic (Resende and Ribeiro 2016) for problem. The proposed modified formulation was capable of solving 38 more instances than the original one. ...
... In Sect. 4, experimental results obtained with the instances proposed in Carrabs et al. (2018b) are presented. Finally, in Sect. ...
Given an edge-colored graph G, a tree with all its edges with different colors is called a rainbow tree. The rainbow spanning forest (RSF) problem consists of finding a spanning forest of G, with the minimum number of rainbow trees. In this paper, we present an integer linear programming model for the RSF problem that improves a previous formulation for this problem. A GRASP metaheuristic is also implemented for providing fast primal bounds for the exact method. Computational experiments carried out over a set of random instances show the effectiveness of the strategies adopted in this work, solving problems in graphs with up to 100 vertices.
... Moreover, in multi-thread computing the power gap increases to 51%. To improve the performances of our genetic algorithm, an idea for future work involves the partitioning of the input graph into subgraphs [10], since we have supposed if it is possible to execute a partition transforming the problem into the rainbow spanning forest problem [2,3]. Considering our preliminary results, we think that this transformation will be the core of our future work concerning this problem. ...
The Minimum Conflict Weighted Spanning Tree Problem is a variant of the Minimum Spanning Tree Problem in which, given a list of conflicting edges modelled as a conflict graph, we want to find a weighted spanning tree with the minimum number of conflicts as main objective function and minimize the total weight of spanning trees as secondary objective function. The problem is proved to be NP-Hard in its general form and finds applications in several real-case scenarios such as the modelling of road networks in which some movements are prohibited. We propose a genetic algorithm designed to minimize the number of conflict edge pairs and the total weight of the spanning tree. We tested our approach on benchmark instances, the results of our GA showed that we outperform the other approaches proposed in the literature.
... In particular, many problems formulated on ELGs have the connectivity as a subjacent objective. This kind of problem has been the subject of research in recent years, as in the works of Carrabs et al. (2017) on the rainbow spanning forest problem, Ismkhan (2017) (refer to Section 4.1), for solving some of these problems. In the sequel, we briefly present five connectivity problems defined on ELGs. ...
The minimum labeling spanning tree problem (MLSTP) is a combinatorial optimization problem that consists in finding a spanning tree in a simple edge-labeled graph, i.e., a graph in which each edge has one label associated, by using a minimum number of labels. It is an NP-hard problem that has attracted substantial research attention in recent years. In its turn, the generalized minimum labeling spanning tree problem (GMLSTP) is a generalization of the MLSTP that allows the situation in which multiple labels can be assigned to an edge. Both problems have several practical applications in important areas such as computer network design, multimodal transportation network design, and data compression. This thesis addresses several connectivity problems defined over edge-labeled graphs, in special the minimum labeling spanning tree problem and its generalized version. The contributions in this work can be classified between theoretical and practical. On the theoretical side, we have introduced new useful concepts, definitions, properties and theorems regarding edge-labeled graphs, as well as a polyhedral study on the GMLSTP. On the practical side, we have proposed new heuristics
— such as the metaheuristic-based algorithm MSLB, and the constructive heuristic MVCA — and exact methods — such as new mathematical formulations and branch-and-cut algorithms — for solving the GMLSTP. Computational experiments over well established benchmarks for the MLSTP are reported, showing that the new approaches introduced in this work have achieved the best results for both heuristic and exact methods in comparison with the state-of-the-art methods in the literature.
... Furthermore, the authors provide a 2-approximation algorithm on trees. Carrabs et al. [5] provide an integer linear programming formulation for the rainbow spanning forest problem, as well as a greedy heuristic algorithm with a multi-start scheme. Jin and Li [14] prove that the problem of finding a set of k monochromatic cycles to cover a graph is NP-hard. ...
... Given the values of the dual variables associated with constraints (2) as u v , ∀ v ∈ V, the pricing subproblem is to find a subgraph c = (V(c), E(c)) such that c ∈ C ⧵ C and ∑ v∈V(c) u v − w c > 0 from (5). Since w c = 1, for a nontrivial cycle and all trivial cycles are already in C, the pricing subproblem is to find the best improving rainbow cycle c that maximizes ∑ v∈V(c) u v − 1. ...
A rainbow cycle in an undirected edge‐colored graph is a cycle in which all edges have different colors. A rainbow cycle cover of a graph is a set of disjoint rainbow cycles, where each vertex belongs to exactly one cycle. The objective of the rainbow cycle cover problem is to minimize the number of rainbow cycles used to cover the vertices of the graph while the trivial cycle version also keeps the number of isolated vertices (called trivial rainbow cycles) at minimum. We present a branch‐and‐price procedure with column generation to solve both versions of the rainbow cycle cover problem. We compare our results with the literature in terms of computational performance. We also discuss two approaches to possibly improve the performance of the branch‐and‐price procedure.
... The use of colors to add another layer of information on graphs has been formalized long ago, in its two main variants -node colored graphs and edge colored graphs, and their power in modeling different types of problems has been extensively discussed (among others, in [6,7,10,12,13]; more specifically, the complexity of finding special paths in edge-colored graphs is discussed in [3]; such problem, as shown later, is deeply connected with the models proposed in this paper, while in [1] the variants generated by graphs with fixed degree is discussed. Additional results for edge-colored paths on multigraph is also discussed in [1]. ...
In this paper we consider a particular graph-optimization problem. Given an edge-colored graph and a set of constraints on the sequence of the colors, one is to find the longest path whose colored edges obey the constraints on the sequence of the colors. In the actual formulation, the problem generalizes already known NP-Complete problems, and, evidently, the alternating path problem in edge colored graphs. Recent literature has shown several contexts where such problem may be useful to model interesting applications, and has proposed exact IP models and related algorithms. We extend on these existing models and extensively test new formulations for the problem, showing how one of the newly developed model clearly exhibits better performance, allowing to solve at optimality instances of significant sizes.
