Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs

The best known method for determining lower bounds on the vertex coloring number of a graph is the linear-programming column-generation technique first employed by Mehrotra and Trick in 1996. We present an implementation of the method that provides numerically safe results, independent of the floating-point accuracy of linearprogramming software. Our work includes an improved branch-and-bound algorithm for maximum-weight stable sets and a parallel branch-and-price framework for graph coloring. Computational results are presented on a collection standard test instances, including the unsolved challenge problems created by David S. Johnson in 1989.

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“…The branch-and-bound algorithm presented here uses depth-first search and adopts ideas from algorithms presented in [3,4,19].…”

Section: Combinatorial Branch and Boundmentioning

confidence: 99%

Safe Lower Bounds for Graph Coloring

Held

Cook

Sewell

2011

Integer Programming and Combinatoral Optimization

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“…The branch-and-bound algorithm presented here uses depth-first search and adopts ideas from algorithms presented in [4,3,22,27]. A subproblem in the branch-and-bound tree consists of a lower bound, denoted LB, which is the weight of the heaviest stable set found so far, the current stable set S = {v 1 , v 2 , .…”

Section: Combinatorial Branch and Boundmentioning

confidence: 99%

“…The first two rules adopt a weighted variation of a technique employed by Balas and Yu [5,4]. Suppose that F ⊆ F and it can be proved that…”

Section: Algorithm 2 Weighted Clique Cover Heuristicmentioning

confidence: 99%

Maximum-weight stable sets and safe lower bounds for graph coloring

2012

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The best method known for determining lower bounds on the vertex coloring number of a graph is the linear-programming column-generation technique, where variables correspond to stable sets, first employed by Mehrotra and Trick in 1996. We present an implementation of the method that provides numerically-safe results, independent of the floating-point accuracy of linear-programming software. Our work includes an improved branch-andbound algorithm for maximum-weight stable sets and a parallel branch-andprice framework for graph coloring. Computational results are presented on a collection of standard test instances, including the unsolved challenge problems created by David S. Johnson in 1989.

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“…Subsequent improvements using B&B [4,17] stimulated development of optimization methods for MWISP [1,2,12,14,16,20,25,28,33,[38][39][40][41]47]. Balas and Xue [3] devised a notable algorithm that extended their previous work [2], using a fast heuristic for the weighted-fractional coloring problem to obtain good upper bounds for B&B.…”

Section: Introductionmentioning

confidence: 98%

A branch-and-price approach for the maximum weight independent set problem

et al. 2005

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The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied problems in combinatorial optimization. This article presents a novel approach to solve MWISP exactly by decomposing the original graph into vertex-induced subgraphs. The approach solves MWISP for the original graph by solving MWISP on the subgraphs to generate columns for a branch-and-price framework. The authors investigate different implementation techniques that can be associated with the approach, and offer computational results to identify the strengths and weaknesses of each implementation technique.

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Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs

Cited by 50 publications

References 6 publications

Safe Lower Bounds for Graph Coloring

Safe Lower Bounds for Graph Coloring

Maximum-weight stable sets and safe lower bounds for graph coloring

A branch-and-price approach for the maximum weight independent set problem

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