Local search for the minimum label spanning tree problem with bounded color classes

Authors:
Tobias BrüGgemann;JéRôMe Monnot;Gerhard J. Woeginger
Affiliations:
Department of Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands;CNRS-LAMSADE, UMR 7024, Université Paris-Dauphine, France;Department of Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Venue:
Operations Research Letters
Year:
2003

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Abstract

In the Minimum Label Spanning Tree problem, the input consists of an edge-colored undirected graph, and the goal is to find a spanning tree with the minimum number of different colors. We investigate the special case where every color appears at most r times in the input graph. This special case is polynomially solvable for r=2, and NP- and APX-complete for any fixed r=3. We analyze local search algorithms that are allowed to switch up to k of the colors used in a feasible solution. We show that for k=2 any local optimum yields an (r+1)/2-approximation of the global optimum, and that this bound is tight. For every k=3, there exist instances for which some local optima are a factor of r/2 away from the global optimum.