89119
15.04.2009

Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems with Applications
Piotr Berman, Marek Karpinski and Andrzej Lingas
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First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomialtime exact solutions. Next, we consider the following general capacitated set cover problem. There is given a set of elements with real weights and a family S of sets of elements. One can use a set if it is a subset of one of the sets on our lists and the sum of weights is at most one. The goal is to cover all the elements with the allowed sets. We show that any polynomialtime algorithm that approximates the uncapacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r + 1.357. In particular, the composition of these two results yields a polynomialtime approximation algorithm for the problem of covering a set of customers represented by a weighted npoint set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. Finally, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs.
