On local search for the generalized graph coloring problem

Authors:
Tjark Vredeveld;Jan Karel Lenstra
Affiliations:
Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O.Box 513, 5600 MB Eindhoven, Netherlands;Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O.Box 513, 5600 MB Eindhoven, Netherlands
Venue:
Operations Research Letters
Year:
2003

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Abstract

Given an edge-weighted graph and an integer k, the generalized graph coloring problem is the problem of partitioning the vertex set into k subsets so as to minimize the total weight of the edges that are included in a single subset. We recall a result on the equivalence between Karush-Kuhn-Tucker points for a quadratic programming formulation and local optima for the simple flip-neighborhood. We also show that the quality of local optima with respect to a large class of neighborhoods may be arbitrarily bad and that some local optima may be hard to find.