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arXiv · 1311.0713

Edge covering with budget constrains

Rajiv Gandhi·G. Kortsarz
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Abstract

We study two related problems: finding a set of k vertices and minimum number of edges (kmin) and finding a graph with at least m' edges and minimum number of vertices (mvms). Goldschmidt and Hochbaum \cite{GH97} show that the mvms problem is NP-hard and they give a 3-approximation algorithm for the problem. We improve \cite{GH97} by giving a ratio of 2. A 2(1+ε)-approximation for the problem follows from the work of Carnes and Shmoys \cite{CS08}. We improve the approximation ratio to 2. algorithm for the problem. We show that the natural LP for \kmin has an integrality gap of 2-o(1). We improve the NP-completeness of \cite{GH97} by proving the pronlem are APX-hard unless a well-known instance of the dense k-subgraph admits a constant ratio. The best approximation guarantee known for this instance of dense k-subgraph is O(n^{2/9}) \cite{BCCFV}. We show that for any constant ρ>1, an approximation guarantee of ρfor the \kmin problem implies a ρ(1+o(1)) approximation for \mwms. Finally, we define we give an exact algorithm for the density version of kmin.

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Rajiv Gandhi, G. Kortsarz. 2013-11-04. Edge covering with budget constrains. https://arxiv.org/abs/1311.0713

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