SETCOVERRNC - SIAM J. COMPUT. Vol. 28, No. 2, pp. 525540 c...

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PRIMAL-DUAL RNC APPROXIMATION ALGORITHMS FOR SET COVER AND COVERING INTEGER PROGRAMS * SRIDHAR RAJAGOPALAN AND VIJAY V. VAZIRANI SIAM J. C OMPUT . c ± 1998 Society for Industrial and Applied Mathematics Vol. 28, No. 2, pp. 525–540 Abstract. We build on the classical greedy sequential set cover algorithm, in the spirit of the primal-dual schema, to obtain simple parallel approximation algorithms for the set cover problem and its generalizations. Our algorithms use randomization, and our randomized voting lemmas may be of independent interest. Fast parallel approximation algorithms were known before for set cover, though not for the generalizations considered in this paper. Key words. algorithms, set cover, primal-dual, parallel, approximation, voting lemmas PII. S0097539793260763 1. Introduction. Given a universe U , containing n elements, and a collection, S · = { S i : S i ⊆U} , of subsets of the universe, the set cover problem asks for the smallest subcollection C⊆S that covers all the n elements in U (i.e., S S C S = U ). In a more general setting, one can associate a cost, c S , with each set S ∈S and ask for the minimum cost subcollection which covers all of the elements. 1 We will use m to denote |S| . Set multicover and multiset multicover are successive natural generalizations of the set cover problem. In both problems, each element e has an integer coverage requirement r e , which specifies how many times e has to be covered. In the case of multiset multicover, element e occurs in a set S with arbitrary multiplicity, denoted m ( S,e ). Setting r e = 1 and choosing m ( ) from { 0 , 1 } to denote whether S contains e gives back the set cover problem. The most general problems we address here are covering integer programs . These are integer programs that have the following form: min c · x , s.t. M x r , x Z + ; the vectors c and r and the matrix M are all nonnegative rational numbers. Because of its generality, wide applicability, and clean combinatorial structure, the set cover problem occupies a central place in the theory of algorithms and approxima- tions. Set cover was one of the problems shown to be NP-complete in Karp’s seminal paper [Ka72]. Soon after this, the natural greedy algorithm, which repeatedly adds the set that contains the largest number of uncovered elements to the cover, was shown to be an H n factor approximation algorithm for this problem ( H n =1+1 / 2+ ... +1 /n ) by Johnson [Jo74] and Lovasz [Lo75]. This result was extended to the minimum cost case by Chvatal [Ch79]. Lovasz establishes a slightly stronger statement, namely, that the ratio of the greedy solution to the optimum fractional solution is at most * Received by the editors November 26, 1993; accepted for publication (in revised form) October 21, 1996; published electronically July 28, 1998. DIMACS, Princeton University, Princeton, NJ 08544. Current address: IBM Almaden Research Center, San Jose, CA 95120 ([email protected]). This research was done while the author was a graduate student at the University of California, Berkeley, supported by NSF PYI Award CCR 88-96202 and NSF grant IRI 91-20074. Part of this work was done while the author was visiting IIT,
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SETCOVERRNC - SIAM J. COMPUT. Vol. 28, No. 2, pp. 525540 c...

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