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Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems

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Abstract

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 (non-necessarily geometric) capacitated set cover problem. There is given a set of elements with real weights and a family of sets of the elements. One can use a set if it is a subset of one of the sets in the family and the sum of the weights of its elements is at most one. The goal is to cover all the elements with the allowed sets.

We show that any polynomial-time 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 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. This substantially improves on the best known approximation ratio for the latter antenna problem equal to 3.

Furthermore, 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.

Finally, we discuss the approximability of the generalization of the antenna problem to include several base stations for antennas, and in particular show its APX-hardness already in the uncapacitated case.

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Notes

  1. 1.

    Very recently, M. Patrascu found a dynamic programming solution of an equivalent problem that has both time and space complexity of O(n 3) [15].

  2. 2.

    Unfortunately, the very recent progress in approximation algorithms for geometric set cover does not seem to include disc sectors which can be arbitrarily thin and thus not even locally fat [1].

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Acknowledgements

The authors are grateful to Martin Wahlen for discussions on MinAntLoad, to David Ilcinkas, Jurek Czyzowicz and Leszek Gasieniec for preliminary discussions on MinAntVar, and to Mihai Patrascu for informing us about his recent time improvement.

Research supported in part by DFG grants, the Hausdorff Center research grant EXC59-1, and the VR grant 621-2008-4649.

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Correspondence to Andrzej Lingas.

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Berman, P., Karpinski, M. & Lingas, A. Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems. Algorithmica 64, 295–310 (2012). https://doi.org/10.1007/s00453-011-9591-5

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Keywords

  • Set cover
  • Geometric set cover
  • Capacitated set cover
  • Assignment of directional antenna
  • Shipping with deadlines
  • Approximation algorithms
  • Polynomial-time approximation scheme
  • APX-hardness
  • Exact algorithms
  • Time complexity