On the complexity of the selective graph coloring problem in some special classes of graphs

Demange, Marc; Monnot, Jérôme; Pop, Petrică C.; Ries, Bernard

doi:10.1016/j.tcs.2013.04.018

Cited by 24 publications

(35 citation statements)

References 15 publications

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“…The inverse chromatic number problem clearly belongs in this class. Another problem of this type is the so-called minimum selective coloring problem (Demange et al, 2015;Demange, Monnot, Petrica, & Ries, 2014). An instance of this problem is a graph whose vertex set is partitioned into p clusters, and the objective is to select one vertex per cluster so as to minimize the chromatic number of the subgraph induced by the selected vertices.…”

Section: Some Close Problemsmentioning

confidence: 99%

Inverse chromatic number problems in interval and permutation graphs

Chung

Culus²,

Demange

2015

European Journal of Operational Research

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Section: Some Close Problemsmentioning

confidence: 99%

Inverse chromatic number problems in interval and permutation graphs

Chung

Culus²,

Demange

2015

European Journal of Operational Research

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“…the number of colors necessary to color the vertices in the selection is minimum. In a graph where each cluster consists of a single vertex, SEL-COL becomes equivalent to the classical graph coloring problem [5]. SEL-COL, or partition coloring as it is alternatively called in the literature, is motivated by the wavelength routing and assignment problem, and was introduced in Li and Simha [17].…”

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“…For each domain of application, they discuss the related model and the complexity of SEL-COL in classes of graphs pointed out by the model. Also, analogies between models as well as new solution approaches and open research directions are highlighted throughout the paper.The selective graph coloring problem is known to be NP-hard, and remains so in many special classes of graphs [5,17]. The work by Li and Simha [17] and that by Noronha and Ribeiro [20] focus on partition coloring in the context of wavelength routing and assignment, and offer heuristic methods to solve the problem.To the best of our knowledge, there are three studies that primarily focus on exact solution methods for SEL-COL.…”

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A decomposition approach to solve the selective graph coloring problem in some perfect graph families

2018

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Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP-hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state-of-the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low-density graphs, and regardless of edge-density in the class of chordal graphs. the number of colors necessary to color the vertices in the selection is minimum. In a graph where each cluster consists of a single vertex, SEL-COL becomes equivalent to the classical graph coloring problem [5]. SEL-COL, or partition coloring as it is alternatively called in the literature, is motivated by the wavelength routing and assignment problem, and was introduced in Li and Simha [17]. There, a telecommunication network, which is a collection of terminal nodes linked by optical fibers capable of carrying a certain number of wavelengths, and a set of source-destination node pairs are given. The problem is concerned with finding the minimum number of distinct wavelengths to assign to each route, that is, paths connecting the given source-destination pairs, such that no two routes that have a common link are assigned the same wavelength. In this setting, the set of all possible routes and pairs of routes possessing a common link correspond to vertices and edges respectively, and groups of routes connecting a given source-destination pair constitute the clusters in the host graph. Then, selection of one vertex per cluster achieves the goal of finding a route between each pair of terminals, and coloring of the selection delivers a proper wavelength to each route.In the standard graph coloring problem, assignments (colors) on the entities (vertices) are done without any regard to alternative choices for them. However, as the example applications reveal, there are cases where entities have their own set of feasible options (clusters) and thus should be allocated one among those alternatives. In such cases, where the basic graph coloring framework fails to suffice, SEL-COL bridges the gap by offering the required flexibility.SEL-COL ha...

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“…Demange et al [10,11] considered this type of graph coloring problem in the framework of generalized network design problems and named it selective graph coloring problem. They investigated as well some special classes of graphs including split graphs, bipartite graphs and q-partite graphs and settled the complexity status of the PGCP in these particular classes.…”

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An improved hybrid ant-local search algorithm for the partition graph coloring problem

Fidanova

Pop

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper we propose a hybrid Ant Colony Optimization (ACO) algorithm for the Partition Graph Coloring Problem (PGCP). Given an undirected graph G = (V , E), whose nodes are partitioned into a given number of the sets, the goal of the PGCP is to find a subset V * ⊂ V that contains exactly one node from each cluster and a coloring for V * so that in the graph induced by V * , two adjacent nodes have different colors and the total number of used colors is minimal. Our hybrid algorithm is obtained by executing a local search procedure after every ACO iteration. The performance of our algorithm is evaluated on a set of instances commonly used as benchmark and the computational results show that compared to state-of-the-art algorithms, our improved hybrid ACO algorithm achieves solid results in very short run-times.

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On the complexity of the selective graph coloring problem in some special classes of graphs

Cited by 24 publications

References 15 publications

Inverse chromatic number problems in interval and permutation graphs

Inverse chromatic number problems in interval and permutation graphs

A decomposition approach to solve the selective graph coloring problem in some perfect graph families

An improved hybrid ant-local search algorithm for the partition graph coloring problem

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