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

Given a hypergraph with nonnegative costs on hyperedges, and a weakly supermodular function r : 2 V→ 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...

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Veröffentlicht in:Discrete Applied Mathematics Jg. 126; H. 2; S. 275 - 289
Hauptverfasser: Zhao, Liang, Nagamochi, Hiroshi, Ibaraki, Toshihide
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Lausanne Elsevier B.V 15.03.2003
Amsterdam Elsevier
New York, NY
Schlagworte:
Approximation algorithm
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph
Graph theory
Hypergraph
Mathematics
Primal dual method
Sciences and techniques of general use
Survivable network design problem
Hypergraph
Graph
Approximation algorithm
Survivable network design problem
Primal dual method
Vertex
Costs
Minimum
Existence theorem
Complete
Functions
Graph theory
Elements
Generalization
Survival network design problem
Survival
Result
Numerical approximation
Design
Maximum
Algorithm performance
Hypermodular function
Network
Problem
Performance
1999
ISSN:0166-218X, 1872-6771
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Beschreibung
Zusammenfassung:Given a hypergraph with nonnegative costs on hyperedges, and a weakly supermodular function r : 2 V→ 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 d max + H(r max ) , where d max + is the maximum degree of hyperedges of positive costs, r max = max S r(S) , and H(k)=1+ 1 2 +⋯+ 1 k . In particular, our result contains a 2 H(r max ) -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).
ISSN:0166-218X
1872-6771
DOI:10.1016/S0166-218X(02)00201-9