## Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems with Applications

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Berman, Piotr

Karpinski, Marek

Lingas, Andrzej

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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 polynomial-time 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.<br>We show that
any polynomial-time algorithm that approximates the un-capacitated 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 polynomial-time approximation
algorithm for the problem of covering a set of customers represented by a
weighted n-point 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.

##### Keywords

Computer Science - Computational Complexity, Computer Science - Discrete Mathematics, Computer Science - Data Structures and Algorithms