Semidefinite Programming Based Algorithms for the Sparsest Cut Problem

Luis A.A. Meira; Flávio K. Miyazawa

RAIRO - Operations Research (2011)

  • Volume: 45, Issue: 2, page 75-100
  • ISSN: 0399-0559

Abstract

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In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all tested instances. The semidefinite relaxation gives a lower bound C W and each heuristic produces a cut S with a ratio c S w S , where either cS is at most a factor of C or wS is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a known semidefinite programming solver.

How to cite

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Meira, Luis A.A., and Miyazawa, Flávio K.. "Semidefinite Programming Based Algorithms for the Sparsest Cut Problem." RAIRO - Operations Research 45.2 (2011): 75-100. <http://eudml.org/doc/276365>.

@article{Meira2011,
abstract = { In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all tested instances. The semidefinite relaxation gives a lower bound $\frac\{C\}\{W\}$ and each heuristic produces a cut S with a ratio $\frac\{c_S\}\{w_S\}$, where either cS is at most a factor of C or wS is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a known semidefinite programming solver. },
author = {Meira, Luis A.A., Miyazawa, Flávio K.},
journal = {RAIRO - Operations Research},
keywords = {Semidefinite programming; Sparsest Cut; combinatorics; semidefinite programming; sparsest cut},
language = {eng},
month = {6},
number = {2},
pages = {75-100},
publisher = {EDP Sciences},
title = {Semidefinite Programming Based Algorithms for the Sparsest Cut Problem},
url = {http://eudml.org/doc/276365},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Meira, Luis A.A.
AU - Miyazawa, Flávio K.
TI - Semidefinite Programming Based Algorithms for the Sparsest Cut Problem
JO - RAIRO - Operations Research
DA - 2011/6//
PB - EDP Sciences
VL - 45
IS - 2
SP - 75
EP - 100
AB - In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all tested instances. The semidefinite relaxation gives a lower bound $\frac{C}{W}$ and each heuristic produces a cut S with a ratio $\frac{c_S}{w_S}$, where either cS is at most a factor of C or wS is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a known semidefinite programming solver.
LA - eng
KW - Semidefinite programming; Sparsest Cut; combinatorics; semidefinite programming; sparsest cut
UR - http://eudml.org/doc/276365
ER -

References

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