Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Abstract

Let D be a set of disks of arbitrary radii in the plane, and let P be a set of points. We study the following three problems: (i) Assuming P contains the set of center points of disks in D , find a minimum-cardinality subset P ∗ of P (if exists), such that each disk in D is pierced by at least h points of P ∗ , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming P is such that for each D ∈ D there exists a point in P whose distance from D's center is at most α r ( D ) , where r ( D ) is D's radius and 0 ⩽ α < 1 is a given constant, find a minimum-cardinality subset P ∗ of P , such that each disk in D is pierced by at least one point of P ∗ . We call this problem minimum discrete piercing with cores. (iii) Assuming P is the set of center points of disks in D , and that each D ∈ D covers at most l points of P , where l is a constant, find a minimum-cardinality subset D ∗ of D , such that each point of P is covered by at least one disk of D ∗ . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a ( 1 + ε ) -approximation for minimum discrete piercing (i.e., for arbitrary P ).