Approximation algorithms for covering polygons with squares and similar problems

Abstract

We consider the problem of covering arbitrary polygons, without any acute interior angles, using a preferably minimum number of squares. The squares must lie entirely within the polygon. Let P be an arbitrary input polygon, with n vertices, coverable by squares. Let μ(P) denote the minimum number of squares required to cover P. In the first part of this paper we present an algorithm which guarantees a constant (14) approximation factor running in O(n ²+μ(P)) time. As a corollary we obtain the first polynomial-time, constant-factor approximation algorithm for “fat” rectangular coverings. In the second part we show an O(n log n+μ(P)) time algorithm which produces at most 11n+μ(P) squares to cover P. In the hole-free case this algorithm runs in linear time and produces a cover which is within an O(α(n)) approximation factor of the optimal, where α(n) is the extremely slowly growing inverse of Ackermann's function. In parallel our algorithm runs in O(log n) randomized time using O(max(μ(P), n)) processors.