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

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 αr(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).