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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Very recently, M. Patrascu found a dynamic programming solution of an equivalent problem that has both time and space complexity of O(n 3) .
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 .
Aronov, B., Ezra, E., Sharir, M.: Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput. 39(7), 3248–3282 (2010)
Bao, L., Garcia-Luna-Aceves, J.: Transmission scheduling in ad hoc networks with directional antennas. In: Proc. ACM MOBICOM, pp. 48–58 (2002)
Berman, P., Kasiviswanathan, S.P., Urgaonkar, B.: Packing to angles and sectors. In: Proc. Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 171–180 (2007)
Berman, P., Karpinski, M.: Improved approximation lower bounds on small occurrence optimization. Electron. Colloq. Computat. Complex. 10(008) (2003)
Broden, B., Hammar, M., Nilsson, B.J.: Guarding lines and 2-link polygons is APX-hard. In: Proc. Canadian Conference on Computational Geometry (CCCG), pp. 45–48 (2001)
Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995)
Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for vertex cover. In: Proc. Annual ACM Symposium on Computational Geometry 2005, pp. 135–141 (2005)
Coffman, E.G., Garey, M.R., Johnson, D.S.: Approximation algorithms for bin packing: a survey. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, pp. 46–93. PWS Publishing, Boston (1997)
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990)
Garey, M.R., Graham, R.L., Johnson, D.S., Yao, A.C.: Resource constrained scheduling as generalized bin-packing. J. Comb. Theory, Ser. A 21, 257–298 (1976), cited from E.G. Coffman, M.R. Garey, D.S. Johnson, Approximation algorithms for bin packing: a survey. In: D.S. Hochbaum (ed.) Approximation Algorithms for NP-hard Problems, p. 50
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-completeness. Freeman, New York (2003)
Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing in image processing and VLSI. J. ACM 32(1), 130–136 (1985)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM 34(1), 144–162 (1987)
Kellerer, H., Pferschy, U., Speranza, M.: An efficient approximation scheme for the subset sum problem. J. Comput. Syst. Sci. 66(2), 349–370 (2003)
Patrascu, M.: Personal communication, August 2009
Peraki, C., Servetto, S.: On the maximum stable throughput problem in random networks with directional antennas. In: Proc. ACM MobiHoc 2003, pp. 76–87 (2003)
Spyropoulos, A., Raghavendra, C.S.: Energy efficient communication in ad hoc networks using directional antennas. In: Proc. IEEE INFOCOM (2002)
Yi, S., Pei, Y., Kalyanaraman, S.: On the capacity improvement of ad hoc wireless networks using directional antennas. In: Proc. ACM MobiHoc, vol. 2003, pp. 108–116 (2003)
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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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
- Set cover
- Geometric set cover
- Capacitated set cover
- Assignment of directional antenna
- Shipping with deadlines
- Approximation algorithms
- Polynomial-time approximation scheme
- Exact algorithms
- Time complexity