Abstract
We give a randomized polynomial time algorithm with approximation ratio O(logφ(n)) for weighted set multi-cover instances with a shallow cell complexity of at most f(n,k) = n φ(n) k O(1). Up to constant factors, this matches a recent result of Könemann et al. for the set cover case, i.e. when all the covering requirements are 1. One consequence of this is an O(1)-approximation for geometric weighted set multi-cover problems when the geometric objects have linear union complexity; for example when the objects are disks, unit cubes or halfspaces in ℝ3. Another consequence is to show that the real difficulty of many natural capacitated set covering problems lies with solving the associated priority cover problem only, and not with the associated multi-cover problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)
Chakrabarty, D., Grant, E., Könemann, J.: On Column-Restricted and Priority Covering Integer Programs. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 355–368. Springer, Heidelberg (2010)
Jain, K.: A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21(1), 39–60 (2001)
Chekuri, C., Clarkson, K.L., Har-Peled, S.: On the set multi-cover problem in geometric settings. In: Symposium on Computational Geometry, pp. 341–350 (2009)
Bansal, N., Pruhs, K.: The geometry of scheduling. In: FOCS, pp. 407–414 (2010)
Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite vc-dimension. Discrete & Computational Geometry 14(4), 463–479 (1995)
Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete & Computational Geometry 37(1), 43–58 (2007)
Varadarajan, K.R.: Epsilon nets and union complexity. In: Symposium on Computational Geometry, pp. 11–16 (2009)
Aronov, B., Ezra, E., Sharir, M.: Small-size epsilon-nets for axis-parallel rectangles and boxes. In: STOC, pp. 639–648 (2009)
Varadarajan, K.: Weighted geometric set cover via quasi-uniform sampling. In: STOC, pp. 641–648 (2010)
Chan, T.M., Grant, E., Könemann, J., Sharpe, M.: Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In: SODA, pp. 1576–1585 (2012)
Alon, N., Spencer, J.: The probabilistic method. John Wiley (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bansal, N., Pruhs, K. (2012). Weighted Geometric Set Multi-cover via Quasi-uniform Sampling. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-33090-2_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33089-6
Online ISBN: 978-3-642-33090-2
eBook Packages: Computer ScienceComputer Science (R0)