# Approximation Algorithms for Rectangle Piercing Problems

Abstract (Summary)

Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our effort to designing approximation algorithms for unit-height rectangles. Our e-approximation scheme requires nO(1/ε²) time. We also consider the problem with restrictions like bounding the depth of a point and the width of the rectangles. The approximation schemes for these two cases take nO(1/ε) time. We also show how to maintain a factor 2 approximation of the piercing set in O(log n) amortized time in an insertion-only scenario.
Bibliographical Information:

Advisor:

School:University of Waterloo

School Location:Canada - Ontario

Source Type:Master's Thesis

Keywords:computer science piercing stabbing algorithm approximation scheme ptas shifting rectangle interval

ISBN:

Date of Publication:01/01/2005