Approximation Algorithms for Partial Covering Problems

We study a generalization of covering problems called partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-partial set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-partial set cover, if each element occurs in at most f sets, then we derive a primal-dual f -approximation algorithm (thus implying a 2-approximation for k-partial vertex cover) in polynomial time. Without making any assumption about the number of sets an element is in, for instances where each set has cardinality at most three, we obtain an approximation of 4/3. We also present better-than-2-approximation algorithms for k-partial vertex cover on bounded degree graphs, and for vertex cover on expanders of bounded average degree. We obtain a polynomial-time approximation scheme for k-partial vertex cover on planar graphs, and for covering k points in Rd by disks.  2004 Elsevier Inc. All rights reserved. ✩ A preliminary version of this work appeared in Proc. International Colloquium on Automata, Languages, and Programming, pp. 225–236, 2001. * Corresponding author. E-mail addresses: [email protected] (R. Gandhi), [email protected] (S. Khuller), [email protected] (A. Srinivasan). 1 Part of this work was done when the author was a student at the University of Maryland and was supported by NSF Award CCR-9820965. 2 Research supported by NSF Award CCR-9820965 and an NSF CCR-0113192. 3 Part of this work was done while at Bell Labs, Lucent Technologies, 600-700 Mountain Avenue, Murray Hill, NJ 07974. Research supported in part by NSF Award CCR-0208005. 0196-6774/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgor.2004.04.002 56 R. Gandhi et al. / Journal of Algorithms 53 (2004) 55–84