Perfectness of clustered graphs

Bonomo, Flavia; Cornaz, Denis; Ekim, Tınaz; Ries, Bernard

doi:10.1016/j.disopt.2013.07.006

Cited by 6 publications

(4 citation statements)

References 12 publications

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“…Theorethical results on the complexity of the PCP on particular classes of graphs have been obtained in Bonomo et al [4], Demange et al [8] and Demange et al [10].…”

Section: Literature Reviewmentioning

confidence: 99%

An exact algorithm for the Partition Coloring Problem

Furini

Malaguti

Santini

2018

Computers & Operations Research

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We study the Partition Coloring Problem (PCP), a generalization of the Vertex Coloring Problem where the vertex set is partitioned. The PCP asks to select one vertex for each subset of the partition in such a way that the chromatic number of the induced graph is minimum. We propose a new Integer Linear Programming formulation with an exponential number of variables. To solve this formulation to optimality, we design an effective Branch-and-Price algorithm. Good quality initial solutions are computed via a new metaheuristic algorithm based on adaptive large neighbourhood search. Extensive computational experiments on a benchmark test of instances from the literature show that our Branch-and-Price algorithm, combined with the new metaheuristic algorithm, is able to solve for the first time to proven optimality several open instances, and compares favourably with the current state-of-the-art exact algorithm.

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“…Theorethical results on the complexity of the PCP on particular classes of graphs have been obtained in Bonomo et al [4], Demange et al [8] and Demange et al [10].…”

Section: Literature Reviewmentioning

confidence: 99%

An exact algorithm for the Partition Coloring Problem

Furini

Malaguti

Santini

2018

Computers & Operations Research

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“…Let us note at that point that unlike in perfect graphs, it is not enough to require these equalities for all induced subgraphs in order to obtain a meaningful definition of selective-perfectness. A formal notion of selective-perfect graphs is introduced and studied in [2].…”

Section: Motivationmentioning

confidence: 99%

On the minimum and maximum selective graph coloring problems in some graph classes

Demange

Ekim

Ries

2016

Discrete Applied Mathematics

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a b s t r a c tGiven a graph together with a partition of its vertex set, the minimum selective coloring problem consists of selecting one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is minimum. The contribution of this paper is twofold. First, we investigate the complexity status of the minimum selective coloring problem in some specific graph classes motivated by some models described in . Second, we introduce a new problem that corresponds to the worst situation in the minimum selective coloring; the maximum selective coloring problem aims to select one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is maximum. We motivate this problem by different models and give some first results concerning its complexity.Input: An undirected graph G = (V , E) and a clustering V = (V 1 , . . . , V p ) of V . Output: A selection V * such that χ (G[V * ]) is minimum. Let k ≥ 1 be a fixed integer. k-Dsel-ColInput: An undirected graph G = (V , E) and a clustering V = (V 1 , . . . , V p ) of V . Question: Does there exist a selection V ′ such that χ (G[V ′ ]) ≤ k? We call such a selection a k-colorable selection. For any k ≥ 1, k-Dsel-Col is clearly in NP in general graphs and consequently we will not mention NP-membership in

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“…Concerning exact algorithms, a branch-and-cut algorithm and a branch-and-price algorithm are given along with computational results in respectively (Frota, Maculan, Noronha, & Ribeiro, 2010;Hoshino, Frota, & de Souza, 2011). More graph theoretical results concerning the selective graph coloring problem can be found in (Bonomo, Cornaz, Ekim, & Ries, 2013).…”

Section: Sel-colmentioning

confidence: 99%

On some applications of the selective graph coloring problem

Demange

Ekim

Ries

et al. 2015

European Journal of Operational Research

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International audienceIn this paper we present the Selective Graph Coloring Problem, a generalization of the standard graph coloring problem as well as several of its possible applications. Given a graph with a partition of its vertex set into several clusters, we want to select one vertex per cluster such that the chromatic number of the subgraph induced by the selected vertices is minimum. This problem appeared in the literature under different names for specific models and its complexity has recently been studied for different classes of graphs. Here, we describe different models – some already discussed in previous papers and some new ones – in very different contexts under a unified framework based on this graph problem. We point out similarities between these models, offering a new approach to solve them, and show some generic situations where the selective graph coloring problem may be used. We focus on specific graph classes motivated by each model, and we briefly discuss the complexity of the selective graph coloring problem in each one of these graph classes and point out interesting future research directions

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Perfectness of clustered graphs

Cited by 6 publications

References 12 publications

An exact algorithm for the Partition Coloring Problem

An exact algorithm for the Partition Coloring Problem

On the minimum and maximum selective graph coloring problems in some graph classes

On some applications of the selective graph coloring problem

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