The Hypocoloring Problem: Complexity and Approximability Results when the Chromatic Number Is Small

Abstract

AbstractWe consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G=(V,E;w) where w(v)≥ 0, the goal consists in finding a partition \({\cal S}=(S_1,\ldots,S_k)\) of the node set of G into hypostable sets and minimizing ∑\(_{i=1}^{k}\) w(S i ) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max { ∑ v ∈ K w(v)| K ∈ S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Δ.KeywordsBipartite GraphMaximum DegreeChromatic NumberTruth AssignmentFree GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.