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A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

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Abstract

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm.

Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k u . A solution consists of a function x:V→ℕ0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x u k u . Our objective is to find a cover that minimizes ∑ v∈V x v w v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time.

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Authors and Affiliations

  1. Department of Computer Science, University of Maryland, College Park, MD, 20742, USA

    Julián Mestre

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  1. Julián Mestre
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Correspondence to Julián Mestre.

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Research supported by NSF Awards CCR 0113192 and CCF 0430650, and the University of Maryland Dean’s Dissertation Fellowship.

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Mestre, J. A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses. Algorithmica 55, 227–239 (2009). https://doi.org/10.1007/s00453-007-9003-z

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  • DOI: https://doi.org/10.1007/s00453-007-9003-z

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