A Branch-and-Cut algorithm for graph coloring

Méndez-Díaz, Isabel; Zabala, Paula

doi:10.1016/j.dam.2005.05.022

Cited by 136 publications

(115 citation statements)

References 19 publications

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“…We have chosen the Dsatur algorithm, since it is used as subroutine in other approaches, e.g., in Méndez-Díaz and Zabala (2006). Table 2 lists all the easy instances along with their number of nodes |V (G)| and edges |E(G)|, density d%, (maximal) clique numberω(G), and chromatic number χ(G) (in italic when it is the best-known value).…”

Section: Problem Instancesmentioning

confidence: 99%

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Gualandi

Malucelli

2012

INFORMS Journal on Computing

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We consider two approaches for solving the classical minimum vertex coloring problem, that is the problem of coloring the vertices of a graph so that adjacent vertices have different colors, minimizing the number of used colors, namely Constraint Programming and Column Generation. Constraint Programming is able to solve very efficiently many of the benchmarks, but suffers from the lack of effective bounding methods.On the contrary, Column Generation provides tight lower bounds by solving the fractional vertex coloring problem exploited in a Branch-and-Price algorithm, as already proposed in the literature. The Column Generation approach is here enhanced by using Constraint Programming to solve the pricing subproblem and to compute heuristic solutions. Moreover new techniques are introduced to improve the performance of the Column Generation approach in solving both the linear relaxation and the integer problem. We report extensive computational results applied to the benchmark instances: we are able to prove optimality of 11 new instances, and to improve the best known lower bounds on other 17 instances. Moreover we extend the solution approaches to a generalization of the problem known as Minimum Vertex Graph Multicoloring Problem where a given number of colors has to be assigned to each vertex.

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Section: Problem Instancesmentioning

confidence: 99%

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Gualandi

Malucelli

2012

INFORMS Journal on Computing

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“…In this section, we present our experimental results, including a comparison between NulLA and other graph coloring algorithms such as the Alon-Tarsi method [1], the Gröb-ner basis method, DSATUR and Branch-and-Cut [12]. To summarize, almost all of the graphs tested by NulLA had degree one certificates.…”

Section: Resultsmentioning

confidence: 99%

“…We conclude with a short comment about NulLA's relation to DSATUR and Branch-and-Cut [12]. These heuristics return bounds on the chromatic number.…”

Section: Resultsmentioning

confidence: 99%

“…These heuristics return bounds on the chromatic number. In Table 5 (data taken from [12]), we display the bounds returned by Branch-and-Cut (B&C) and DSATUR, respectively. In the case of these graphs, NulLA determined non-3-colorability very rapidly (establishing a lower bound of four), while the two heuristics returned lower bounds of three and two, respectively.…”

Section: Resultsmentioning

confidence: 99%

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Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

Loera

Lee²,

Malkin

et al. 2008

Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation

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Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.

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“…Since then, a considerable number of new techniques have been developed and important progress has been made. While exact approaches have been tried on graph coloring [Brelaz, 1979;Mehrotra and Trick, 1996;Sewell, 1996;Ramani et al, 2006;Lucet et al, 2006;Méndez-Díaz and Zabala, 2006], "very few exact solution algorithms exist for the problem" [Méndez-Díaz and Zabala, 2006]. As such, heuristic methods dominate the literature of practical algorithms of general graph coloring.…”

Section: Existing Heuristic Approachesmentioning

confidence: 99%

Heuristic algorithms and learning techniques: applications to the graph coloring problem

Porumbel¹

2011

4OR-Q J Oper Res

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A Branch-and-Cut algorithm for graph coloring

Cited by 136 publications

References 19 publications

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

Heuristic algorithms and learning techniques: applications to the graph coloring problem

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