NP-completeness and FPT Results for Rectilinear Covering Problems

Abstract: This paper discusses three rectilinear (that is, axis-parallel) covering problems in d dimensions and their variants. The first problem is the Rectilinear Line Cover where the inputs are n points in Rd and a positive integer k, and we are asked to answer if we can cover these n points with at most k lines where these lines are restricted to be axis parallel. We show that this problem has efficient fixed-parameter tractable (FPT) algorithms. The second problem is the Rectilinear k-Links Spanning Path Problem where the inputs are also n points in Rd and a positive integer k but here we are asked to answer if there is a piecewise linear path through these n points having at most k line-segments (links) where these line-segments are axisparallel. We prove that this second problem is FPT under the assumption that no two line-segments share the same line. The third problem is the Rectilinear Hyperplane Cover problem and we are asked to cover a set of n points in d dimensions with...

• Type: Journal
• Year: 2010
• Where: JUCS
• Authors: Vladimir Estivill-Castro, Apichat Heednacram, Francis Suraweera
• Keywords: JUCS 2010, Path Problem, Rectilinear Hyperplane Cover, Rectilinear Minimum Link