NP-completeness of spreading colored points

Authors:
Ovidiu Daescu;Wenqi Ju;Jun Luo
Affiliations:
Department of Computer Science, University of Texas at Dallas;Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, China and Institute of Computing Technology, Chinese Academy of Sciences, China;Institute of Computing Technology, Chinese Academy of Sciences, China
Venue:
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
Year:
2010

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Abstract

There are n points in the plane and each point is painted by one of m colors where m ≤ n. We want to select m different color points such that (1) the total edge length of resulting minimal spanning tree is as small as possible; or (2) the total edge length of resulting minimal spanning tree is as large as possible; or (3) the perimeter of the convex hull of m different color points is as small as possible. We prove NP-completeness for those three problems and give approximations algorithms for the third problem.