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Pruning 2-Connected Graphs

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Abstract

Given an undirected graph G with edge costs and a specified set of terminals, let the density of any subgraph be the ratio of its cost to the number of terminals it contains. If G is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of G? We answer this question in the affirmative by giving an algorithm to pruneG and find such subgraphs of any desired size, incurring only a logarithmic factor increase in density (plus a small additive term).

We apply our pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. In the Budget-2VC problem, we are given a graph G with edge costs, and a budget B; the goal is to find a 2-vertex-connected subgraph H of G with total edge cost at most B that maximizes the number of vertices in H. We describe an O(log nlog k) approximation for the k-2VC problem, and a bicriteria approximation for the Budget-2VC problem that gives an \(O(\frac{1}{\epsilon}\log^{2} n)\) approximation, while violating the budget by a factor of at most 2+ε.

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Authors and Affiliations

  1. Dept. of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Chandra Chekuri

  2. Google Research, 76 Ninth Ave, 4th Floor, New York, NY, 10011, USA

    Nitish Korula

Authors
  1. Chandra Chekuri
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  2. Nitish Korula
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Correspondence to Nitish Korula.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 28th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2008.

C. Chekuri was partially supported by NSF grants CCF 0728782 and CNS-0721899, and a US-Israeli BSF grant 2002276.

When this work was done, N. Korula was at the Department of Computer Science at the University of Illinios, and was partially supported by NSF grant CCF 0728782.

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Chekuri, C., Korula, N. Pruning 2-Connected Graphs. Algorithmica 62, 436–463 (2012). https://doi.org/10.1007/s00453-010-9462-5

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