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## Coloring Problem (coloring + problem)
## Selected Abstracts## A survey on vertex coloring problems INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 1 2010Enrico MalagutiAbstract This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported. [source] ## ChemInform Abstract: An Application of the "Coloring Problem": Structure,Composition,Bonding Relationships in the Magnetocaloric Materials LaFe13-xSix. CHEMINFORM, Issue 13 2008Mi-Kyung HanAbstract ChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 200 leading journals. To access a ChemInform Abstract of an article which was published elsewhere, please select a "Full Text" option. The original article is trackable via the "References" option. [source] ## Disjoint clique cutsets in graphs without long holes JOURNAL OF GRAPH THEORY, Issue 4 2005Elaine M. EschenAbstract A biclique cutset is a cutset that induces the disjoint union of two cliques. A hole is an induced cycle with at least five vertices. A graph is biclique separable if it has no holes and each induced subgraph that is not a clique contains a clique cutset or a biclique cutset. The class of biclique separable graphs contains several well-studied graph classes, including triangulated graphs. We prove that for the class of biclique separable graphs, the recognition problem, the vertex coloring problem, and the clique problem can be solved efficiently. Our algorithms also yield a proof that biclique separable graphs are perfect. Our coloring algorithm is actually more general and can be applied to graphs that can be decomposed via a special type of biclique cutset. Our algorithms are based on structural results on biclique separable graphs developed in this paper. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 277,298, 2005 [source] ## Improved bounds for the chromatic number of a graph JOURNAL OF GRAPH THEORY, Issue 3 2004S. Louis HakimiAbstract After giving a new proof of a well-known theorem of Dirac on critical graphs, we discuss the elegant upper bounds of Matula and Szekeres-Wilf which follow from it. In order to improve these bounds, we consider the following fundamental coloring problem: given an edge-cut (V1, V2) in a graph G, together with colorings of ,V1, and ,V2,, what is the least number of colors in a coloring of G which "respects" the colorings of ,V1, and ,V2, ? We give a constructive optimal solution of this problem, and use it to derive a new upper bound for the chromatic number of a graph. As easy corollaries, we obtain several interesting bounds which also appear to be new, as well as classical bounds of Dirac and Ore, and the above mentioned bounds of Matula and Szekeres-Wilf. We conclude by considering two algorithms suggested by our results. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 217,225, 2004 [source] ## A branch-and-cut algorithm for partition coloring NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2010Yuri FrotaAbstract Let G = (V, E, Q) be a undirected graph, where V is the set of vertices, E is the set of edges, and Q = {Q1,,,Qq} is a partition of V into q subsets. We refer to Q1,,,Qq as the components of the partition. The partition coloring problem (PCP) consists of finding a subset V, of V with exactly one vertex from each component Q1,,,Qq and such that the chromatic number of the graph induced in G by V, is minimum. This problem is a generalization of the graph coloring problem. This work presents a branch-and-cut algorithm proposed for PCP. An integer programing formulation and valid inequalities are proposed. A tabu search heuristic is used for providing primal bounds. Computational experiments are reported for random graphs and for PCP instances originating from the problem of routing and wavelength assignment in all-optical WDM networks. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## On the complexity of resilient network design NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2010Artur TomaszewskiAbstract In this article we prove ,,,,-hardness of two well-known optimization problems related to the design of multicommodity flow networks with two different methods for providing network resiliency against failures: path diversity and flow restoration. Path diversity is a static mechanism that consists of using, for each demand, a number of paths and oversizing the flows assigned to these paths so that for any failure the total surviving flow is not less than the volume of the demand. By contrast, flow restoration is a dynamic mechanism that consists of reassigning the failed flows to backup paths when a failure occurs. Both mechanisms are of practical interest because although flow restoration is in general superior to path diversity in terms of the required amount of resource capacity, it might be too complicated to implement. By providing an appropriate reduction from the fractional graph coloring problem, we show that both problems are ,,,,-hard in the general case of failure scenarios that admit simultaneous failures of multiple links. Finally, we discuss how to efficiently solve the two problems using path generation techniques. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## Improper coloring of unit disk graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2009Frédéric HavetAbstract Motivated by a satellite communications problem, we consider a generalized coloring problem on unit disk graphs. A coloring is k -improper if no more than k neighbors of every vertex have the same colour as that assigned to the vertex. The k -improper chromatic number ,k(G) is the least number of colors needed in a k -improper coloring of a graph G. The main subject of this work is analyzing the complexity of computing ,k for the class of unit disk graphs and some related classes, e.g., hexagonal graphs and interval graphs. We show NP-completeness in many restricted cases and also provide both positive and negative approximability results. Because of the challenging nature of this topic, many seemingly simple questions remain: for example, it remains open to determine the complexity of computing ,k for unit interval graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009 [source] ## Approximate L(,1,,2,,,,t)-coloring of trees and interval graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Alan A. BertossiAbstract Given a vector (,1,,2,,,,t) of nonincreasing positive integers, and an undirected graph G = (V,E), an L(,1,,2,,,,t)-coloring of G is a function f from the vertex set V to a set of nonnegative integers such that ,f(u) , f(v), , ,i, if d(u,v) = i, 1 , i , t, where d(u,v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(,1,,2,,,,t)-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(,1,,2,,,,t)-coloring of two relevant classes of graphs,trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t + ,1)) and O(nt2,1) time algorithms are proposed to find ,-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and , is a constant depending on t and ,1,,,,t. Moreover, an O(n) time algorithm is given for the L(,1,,2)-coloring of unit interval graphs, which provides a 3-approximation. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 204,216 2007 [source] ## Solving the minimum-weighted coloring problem NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2001Massimiliano CaramiaAbstract Weighted coloring is a generalization of the well-known vertex (unweighted) coloring for which a number of exact algorithms have been presented in the literature. We are not aware of any optimal method specifically designed for the minimum-weighted coloring problem on arbitrary graphs. Only a few heuristics have been developed with the goal of finding tighter upper bounds for the maximum-weighted clique problem. Moreover, as shown in the paper, a straightforward reduction of a weighted instance into an unweighted one permits us to solve only very small instances. In this paper, we present a branch-and-bound algorithm for the weighted case capable of solving random graphs of up to 90 vertices for any edge density with integer weights uniformly drawn from the range [1, ,,10]. Likewise, we have used properly modified benchmark instances borrowed from vertex coloring as a further test bed for our algorithm. © 2001 John Wiley & Sons, Inc. [source] ## A survey on vertex coloring problems INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 1 2010Enrico MalagutiAbstract This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported. [source] |