Geometric Minimum Diameter Minimum Cost Spanning Tree Problem

Abstract

In this paper we consider bi-criteria geometric optimization problems, in particular, the minimum diameter minimum cost spanning tree problem and the minimum radius minimum cost spanning tree problem for a set of points in the plane. The former problem is to construct a minimum diameter spanning tree among all possible minimum cost spanning trees, while the latter is to construct a minimum radius spanning tree among all possible minimum cost spanning trees. The graph-theoretic minimum diameter minimum cost spanning tree (MDMCST) problem and the minimum radius minimum cost spanning tree (MRMCST) problem have been shown to be NP-hard. We will show that the geometric version of these two problems, GMDMCST problem and GMRMCST problem are also NP-hard. We also give two heuristic algorithms, one MCST-based and the other MDST-based for the GMDMCST problem and present some experimental results.

Research supported in part by the National Science Council under the Grants No. NSC-94-2213-E-001-004, NSC-95-2221-E-001-016-MY3, and NSC 94-2752-E-002-005-PAE, and by the Taiwan Information Security Center (TWISC), National Science Council under the Grant No. NSC94-3114-P-001-001-Y.