Abstract
We study the partial vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G=(V,E) and an integer s, the 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 vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(V log V + E) time. This represents an improvement in running time from the previously known fastest algorithm.
Our technique can also be applied to a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k u and a weight w u . A solution consists of a function x: V →ℕ0 and an orientation of all but s edges, such that the number edges oriented toward any vertex u is at most x u k u . The cost of the cover is given by ∑ v ∈ V x v w v . Our objective is to find a cover with minimum cost. We provide an algorithm with the same performance guarantee as for regular partial vertex cover. In this case no algorithm for the problem was known.
Research supported by NSF Awards CCR 0113192 and CCF 0430650
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Mestre, J. (2005). A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_16
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DOI: https://doi.org/10.1007/11538462_16
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