A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem

doi:10.1016/S0304-0208(08)73101-3

North-Holland Mathematics Studies

Volume 109, 1985, Pages 27-45

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A local-ratio theorem for approximating the weighted vertex cover problem is presented. It consists of reducing the weights of vertices in certain subgraphs and has the effect of local-approximation.

Putting together the Nemhauser-Trotter local optimization algorithm and the local-ratio theorem yields several new approximation techniques which improve known results from time complexity, simplicity and performance-ratio point of view.

The main approximation algorithm guarantees a ratio of

where K is the smallest integer s.t.

^†

This is an improvement over the currently known ratios, especially for a “practical” number of vertices (e.g. for graphs which have less than 2400, 60000, 10¹² vertices the ratio is bounded by 1.75, 1.8, 1.9 respectively).

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^†: All log bases, in this paper, are 2.

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A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem

J. of Algorithms

Dis. App. Math.

A Greedy-Heuristic for the Set-Covering Problem

Math. Operations Res.

Applications of the Upton and Tarjan's Planar Separator Theorem

J Information Proc.

On the Problem of Partitioning Planar graphs

SIAM J. on Alg. and Disc. Meth.

Graph Algorithms

Computers an Intractability; A Guide to the Theory of NP-Completeness