CirCut
Rank-two relaxation heuristics for MAX-CUT and other binary quadratic programs The Goemans--Williamson randomized algorithm guarantees a high-quality approximation to the MAX-CUT problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized version of the Goemans--Williamson technique. The proposed approach leads to continuous optimization heuristics applicable to MAX-CUT as well as other binary quadratic programs, for example the MAX-BISECTION problem.par A computer code based on the rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes that implement the Goemans--Williamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular MAX-CUT problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the Goemans--Williamson randomized algorithm. An extension to MAX-BISECTION is also discussed, as is an important difference between the proposed approach and the Goemans--Williamson algorithm; namely, that the new approach does not guarantee an upper bound on the MAX-CUT optimal value.
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References in zbMATH (referenced in 35 articles , 1 standard article )
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Sorted by year (- Mu, Xuewen; Liu, Wenlong: An augmented Lagrangian method for binary quadratic programming based on a class of continuous functions (2016)
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- Ling, Ai-fan; Xu, Cheng-xian: A new discrete filled function method for solving large scale max-cut problems (2012)
- Shylo, V.P.; Shylo, O.V.; Roschyn, V.À.: Solving weighted max-cut problem by global equilibrium search (2012)
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- Wu, Z.Y.; Li, G.Q.; Quan, J.: Global optimality conditions and optimization methods for quadratic integer programming problems (2011)
- Fan, Neng; Pardalos, Panos M.: Linear and quadratic programming approaches for the general graph partitioning problem (2010)
- Kisialiou, Mikalai; Luo, Zhi-Quan: Probabilistic analysis of semidefinite relaxation for binary quadratic minimization (2010)
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- Shylo, V.P.; Shylo, O.V.: Solving the maxcut problem by the global equilibrium search (2010)
- Xia, Yong; Xu, Zi: An efficient Lagrangian smoothing heuristic for max-cut (2010)
- Laguna, Manuel; Duarte, Abraham; Martí, Rafael: Hybridizing the cross-entropy method: An application to the max-cut problem (2009)
- Ling, Ai-Fan; Xu, Cheng-Xian; Xu, Feng-Min: A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems (2009)
- Boros, Endre; Hammer, Peter L.; Sun, Richard; Tavares, Gabriel: A Max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO) (2008)
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- Pan, Shaohua; Tan, Tao; Jiang, Yuxi: A global continuation algorithm for solving binary quadratic programming problems (2008)
- Śliwa, I.; Grygiel, K.; Szlachetka, P.: Hyperchaotic beats and their collapse to the quasiperiodic oscillations (2008)
- Burer, Samuel; Lee, Jon: Solving maximum-entropy sampling problems using factored masks (2007)