论文信息 - A 4.31-approximation for the geometric unique coverage problem on unit disks

A 4.31-approximation for the geometric unique coverage problem on unit disks

We give an improved approximation algorithm for the unique unit-disk coverage problem: Given a set of points and a set of unit disks, both in the plane, we wish to find a subset of disks that maximizes the number of points contained in exactly one disk in the subset. Erlebach and van Leeuwen (2008) introduced this problem as the geometric version of the unique coverage problem, and gave a polynomial-time 18-approximation algorithm. In this paper, we improve this approximation ratio 18 to 2+4/[email protected]( 0. Our algorithm runs in polynomial time which depends exponentially on 1/@e. The algorithm can be generalized to the budgeted unique unit-disk coverage problem in which each point has a profit, each disk 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.

[1] Chak-Kuen Wong,et al. Approximate Algorithms for Some Generalized Knapsack Problems , 1976, Theor. Comput. Sci..

[2] Wolfgang Maass,et al. Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[3] van Leeuwen,et al. Optimization and approximation on systems of geometric objects , 2009 .

[4] Brenda S. Baker,et al. Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[5] Vijay V. Vazirani,et al. Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[6] Venkatesan Guruswami,et al. The Complexity of Making Unique Choices: Approximating 1-in- k SAT , 2005, APPROX-RANDOM.

[7] Yota Otachi,et al. A 4.31-Approximation for the Geometric Unique Coverage Problem on Unit Disks , 2012, ISAAC.

[8] Erik Jan van Leeuwen,et al. Approximating geometric coverage problems , 2008, SODA '08.

[9] Hannes Moser,et al. The Parameterized Complexity of Unique Coverage and Its Variants , 2011, Algorithmica.

[10] Yota Otachi,et al. A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares , 2016, Comput. Geom..

[11] U. Feige,et al. Combination can be hard: approximability of the unique coverage problem , 2006, SODA 2006.