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Improved semidefinite bounding procedure for solving Max-Cut problems to optimality

Nathan Krislock 1, * Jérôme Malick 1 Frédéric Roupin 2 
* Corresponding author
1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : We present an improved algorithm for finding exact solutions to Max-Cut and the related binary quadratic programming problem, both classic problems of combinatorial optimization. The algorithm uses a branch-(and-cut-)and-bound paradigm, using standard valid inequalities and nonstandard semidefinite bounds. More specifically, we add a quadratic regularization term to the strengthened semidefinite relaxation in order to use a quasi-Newton method to compute the bounds. The ratio of the tightness of the bounds to the time required to compute them can be controlled by two real parameters; we show how adjusting these parameters and the set of strengthening inequalities gives us a very efficient bounding procedure. Embedding our bounding procedure in a generic branch-and-bound platform, we get a competitive algorithm: extensive experiments show that our algorithm dominates the best existing method.
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Submitted on : Friday, February 3, 2012 - 12:06:35 PM
Last modification on : Thursday, January 20, 2022 - 5:28:36 PM
Long-term archiving on: : Thursday, November 22, 2012 - 10:40:29 AM

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Nathan Krislock, Jérôme Malick, Frédéric Roupin. Improved semidefinite bounding procedure for solving Max-Cut problems to optimality. Mathematical Programming, 2014, 143 (1-2), pp.61-86. ⟨10.1007/s10107-012-0594-z⟩. ⟨hal-00665968⟩

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