A primal-dual approximation algorithm for the survivable network design problem in hypergraphs

  • Authors:

  • Liang Zhao;Hiroshi Nagamochi;Toshihide Ibaraki

  • Affiliations:

  • Graduate School of Informatics, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, 606-8501, Japan;Toyohashi University of Technology, Department of Information and Computer Sciences, Toyohashi, Aichi, 441-8580, Japan;Graduate School of Informatics, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, 606-8501, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2003

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Abstract

Given a hypergraph with nonnegative costs on hyperedges, and a weakly supermodular function r:2V → Z+, where V is the vertex set, we consider the problem of finding a minimum cost subset of hyperedges such that for every set S ⊆ V, there are at least r(S) hyperedges that have at least one but no all endpoints in S. This problem captures a hypergraph generalization of the survivable network design problem (SNDP), and also the element connectivity problem (ECP). We present a primal-dual algorithm with a performance guarantee of dmax+H (rmax), where dmax+ is the maximum degree of hyperedges of positive costs, rmax = maxs r(S), and H(k) = 1 + 1/2 +...+ 1/k. In particular, our result contains a 2H(rmax)-approximation algorithm for ECP, which gives an independent and complete proof for the result first obtained by Jain et al. (Proceedings of the SODA, 1999, p. 484-489).