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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| Published in: | Discrete Applied Mathematics Vol. 126; no. 2; pp. 275 - 289 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Lausanne
Elsevier B.V
15.03.2003
Amsterdam Elsevier New York, NY |
| Subjects: | |
| ISSN: | 0166-218X, 1872-6771 |
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| Abstract | 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). |
|---|---|
| AbstractList | 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). |
| Author | Nagamochi, Hiroshi Zhao, Liang Ibaraki, Toshihide |
| Author_xml | – sequence: 1 givenname: Liang surname: Zhao fullname: Zhao, Liang email: zhao@amp.i.kyoto-u.ac.jp organization: Graduate School of Informatics, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, 606-8501, Japan – sequence: 2 givenname: Hiroshi surname: Nagamochi fullname: Nagamochi, Hiroshi email: naga@ics.tut.ac.jp organization: Toyohashi University of Technology, Department of Information and Computer Sciences, Toyohashi, Aichi, 441-8580, Japan – sequence: 3 givenname: Toshihide surname: Ibaraki fullname: Ibaraki, Toshihide email: ibaraki@amp.i.kyoto-u.ac.jp organization: Graduate School of Informatics, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, 606-8501, Japan |
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| Keywords | 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 |
| Language | English |
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| References_xml | – volume: 8 start-page: 399 year: 1956 end-page: 404 ident: BIB2 article-title: Maximal flow through a network publication-title: Canad. J. Math. – reference: D.P. Williamson, The primal–dual method for approximation algorithms, Math. Program., to appear. – volume: 15 start-page: 435 year: 1995 end-page: 454 ident: BIB14 article-title: A primal–dual approximation algorithm for generalized Steiner network problems publication-title: Combinatorica – reference: L. Zhao, H. Nagamochi, T. Ibaraki, A note on approximating the survivable network design problem in hypergraphs, IEICE Trans. Inform. Systems E85-D (2) (2002), 322–326. – year: 1998 ident: BIB9 publication-title: Multicast Networking and Applications – reference: R. Ravi, D.P. Williamson, Erratum: an approximation algorithm for minimum-cost vertex-connectivity problems, manuscript, July 2001. – reference: L. Fleischer, K. Jain, D.P. Williamson, An iterative rounding 2-approximation algorithms for the element connectivity problem, in: Proceedings of FOCS 2001. – start-page: 144 year: 1997 end-page: 191 ident: BIB4 article-title: The primal–dual method for approximation algorithms and its application to network design problems publication-title: Approximation Algorithms for NP-hard Problems, PWS – reference: K. Jain, I. Măndoiu, V.V. Vazirani, D.P. Williamson, A primal–dual schema based approximation algorithm for the element connectivity problem, in: Proceedings of the SODA 1999, pp. 484–489. – volume: 35 start-page: 921 year: 1988 end-page: 940 ident: BIB6 article-title: A new approach to the maximum flow problem publication-title: J. ACM – volume: 18 start-page: 21 year: 1997 end-page: 43 ident: BIB10 article-title: An approximation algorithm for minimum-cost vertex-connectivity problems publication-title: Algorithmica – volume: 24 start-page: 296 year: 1995 end-page: 317 ident: BIB3 article-title: A general approximation technique for constrained forest problems publication-title: SIAM J. Comput. – reference: K. Takeshita, T. Fujito, T. Watanabe, On primal–dual approximation algorithms for several hypergraph problems, IPSJ Math. Model. Problem Solving 23 (3) (1999) 13–18 (in Japanese). – volume: 21 start-page: 39 year: 2001 end-page: 60 ident: BIB7 article-title: A factor 2 approximation algorithm for the generalized Steiner network problem publication-title: Combinatorica – reference: M.X. Goemans, A.V. Goldberg, S. Plotkin, D. Shmoys, E. Tardos, D.P. Williamson, Improved approximation algorithms for network design problems, in: Proceedings of the SODA 1994, pp. 223–232. – volume: 21 start-page: 39 issue: 1 year: 2001 ident: 10.1016/S0166-218X(02)00201-9_BIB7 article-title: A factor 2 approximation algorithm for the generalized Steiner network problem publication-title: Combinatorica doi: 10.1007/s004930170004 – ident: 10.1016/S0166-218X(02)00201-9_BIB8 – year: 1998 ident: 10.1016/S0166-218X(02)00201-9_BIB9 – start-page: 144 year: 1997 ident: 10.1016/S0166-218X(02)00201-9_BIB4 article-title: The primal–dual method for approximation algorithms and its application to network design problems – ident: 10.1016/S0166-218X(02)00201-9_BIB12 – volume: 15 start-page: 435 issue: 3 year: 1995 ident: 10.1016/S0166-218X(02)00201-9_BIB14 article-title: A primal–dual approximation algorithm for generalized Steiner network problems publication-title: Combinatorica doi: 10.1007/BF01299747 – ident: 10.1016/S0166-218X(02)00201-9_BIB15 – ident: 10.1016/S0166-218X(02)00201-9_BIB11 doi: 10.1007/s00453-002-0970-9 – ident: 10.1016/S0166-218X(02)00201-9_BIB1 doi: 10.1109/SFCS.2001.959908 – ident: 10.1016/S0166-218X(02)00201-9_BIB13 – volume: 18 start-page: 21 year: 1997 ident: 10.1016/S0166-218X(02)00201-9_BIB10 article-title: An approximation algorithm for minimum-cost vertex-connectivity problems publication-title: Algorithmica doi: 10.1007/BF02523686 – ident: 10.1016/S0166-218X(02)00201-9_BIB5 – volume: 35 start-page: 921 year: 1988 ident: 10.1016/S0166-218X(02)00201-9_BIB6 article-title: A new approach to the maximum flow problem publication-title: J. ACM doi: 10.1145/48014.61051 – volume: 24 start-page: 296 issue: 2 year: 1995 ident: 10.1016/S0166-218X(02)00201-9_BIB3 article-title: A general approximation technique for constrained forest problems publication-title: SIAM J. Comput. doi: 10.1137/S0097539793242618 – volume: 8 start-page: 399 year: 1956 ident: 10.1016/S0166-218X(02)00201-9_BIB2 article-title: Maximal flow through a network publication-title: Canad. J. Math. doi: 10.4153/CJM-1956-045-5 |
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| Snippet | 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... |
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| StartPage | 275 |
| SubjectTerms | 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 |
| Title | A primal–dual approximation algorithm for the survivable network design problem in hypergraphs |
| URI | https://dx.doi.org/10.1016/S0166-218X(02)00201-9 |
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