A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares

Authors:
Takehiro Ito;Shin-Ichi Nakano;Yoshio Okamoto;Yota Otachi;Ryuhei Uehara;Takeaki Uno;Yushi Uno
Affiliations:
Graduate School of Information Sciences, Tohoku University, Japan;Department of Computer Science, Gunma University, Japan;Department of Communication Engineering and Informatics, University of Electro-Communications, Japan;School of Information Science, JAIST, Japan;School of Information Science, JAIST, Japan;Principles of Informatics Research Division, National Institute of Informatics, Japan;Graduate School of Science, Osaka Prefecture University, Japan
Venue:
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Year:
2012

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Abstract

We give a polynomial-time approximation scheme for the unique unit-square coverage problem: given a set of points and a set of axis-parallel unit squares, both in the plane, we wish to find a subset of squares that maximizes the number of points contained in exactly one square in the subset. Erlebach and van Leeuwen (2008) introduced this problem as the geometric version of the unique coverage problem, and the best approximation ratio by van Leeuwen (2009) before our work was 2. Our scheme can be generalized to the budgeted unique unit-square coverage problem, in which each point has a profit, each square has a cost, and we wish to maximize the total profit of the uniquely covered points under the condition that the total cost is at most a given bound.