Lower Bound for (Sum) Coloring Problem
Alexandre Gondran (ENAC), Vincent Duchamp (ENAC), Laurent Moalic, (Universit\'e de Haute-Alsace (UHA))
TL;DR
This paper introduces a novel lower bound technique for the Minimum Sum Coloring Problem, improving bounds on benchmark graphs and establishing optimal values for some instances.
Contribution
It proposes a new relaxation approach transforming the problem into an integer partition with constraints, enhancing lower bounds and confirming optimal solutions.
Findings
Improved lower bounds for 18 benchmark graphs
Proved optimal values for 4 graphs
Demonstrated effectiveness of the relaxation method
Abstract
The Minimum Sum Coloring Problem is a variant of the Graph Vertex Coloring Problem, for which each color has a weight. This paper presents a new way to find a lower bound of this problem, based on a relaxation into an integer partition problem with additional constraints. We improve the lower bound for 18 graphs of standard benchmark DIMACS, and prove the optimal value for 4 graphs by reaching their known upper bound.
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Taxonomy
TopicsScheduling and Timetabling Solutions · graph theory and CDMA systems · Advanced Optimization Algorithms Research