Local Search Algorithms for the k-cardinality Tree Problem

In this paper we deal with an NP-hard combinatorial optimization problem, the k-cardinality tree problem in node-weighted graphs. This problem has several applications, which justify the need for efficient methods to obtain good solutions. We review existing literature on the problem. Then we prove that under the condition that the graph contains exactly one trough, the problem can be solved in polynomial time. For the general NP-hard problem we implemented several local search methods to obtain heuristic solutions, which are qualitatively better than solutions found by constructive heuristics and which require significantly less time than needed to obtain optimal solutions. We used the well-known concepts of genetic algorithms and tabu search with useful extensions. The general performance of our methods is illustrated by numerical results.

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