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Local search for the minimum label spanning tree problem with bounded color classes. (English) Zbl 1046.90070

Summary: 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 r3.

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 k3, there exist instances for which some local optima are a factor of r/2 away from the global optimum.

MSC:
90C27Combinatorial optimization
90C59Approximation methods and heuristics
90C40Markov and semi-Markov decision processes
05C85Graph algorithms (graph theory)