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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 $r⩾3$.

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 $\left(r+1\right)/2$-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.

##### MSC:
 90C27 Combinatorial optimization 90C59 Approximation methods and heuristics 90C40 Markov and semi-Markov decision processes 05C85 Graph algorithms (graph theory)