A Semidefinite Programming Relaxation for the Generalized Stable Set Problem
Summary :
In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E88-A No.5 pp.1122-1128
- Publication Date
- 2005/05/01
- Publicized
- Online ISSN
- DOI
- 10.1093/ietfec/e88-a.5.1122
- Type of Manuscript
- Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
- Category
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Tetsuya FUJIE, Akihisa TAMURA, "A Semidefinite Programming Relaxation for the Generalized Stable Set Problem" in IEICE TRANSACTIONS on Fundamentals,
vol. E88-A, no. 5, pp. 1122-1128, May 2005, doi: 10.1093/ietfec/e88-a.5.1122.
Abstract: In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e88-a.5.1122/_p
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@ARTICLE{e88-a_5_1122,
author={Tetsuya FUJIE, Akihisa TAMURA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Semidefinite Programming Relaxation for the Generalized Stable Set Problem},
year={2005},
volume={E88-A},
number={5},
pages={1122-1128},
abstract={In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.},
keywords={},
doi={10.1093/ietfec/e88-a.5.1122},
ISSN={},
month={May},}
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TY - JOUR
TI - A Semidefinite Programming Relaxation for the Generalized Stable Set Problem
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1122
EP - 1128
AU - Tetsuya FUJIE
AU - Akihisa TAMURA
PY - 2005
DO - 10.1093/ietfec/e88-a.5.1122
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E88-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 2005
AB - In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.
ER -
English
