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Linear size MIP formulation of Max-Cut: new properties, links with cycle inequalities and computational results

Abstract : We consider the Max-Cut problem on an undirected graph G = (V, E) with |V | = n nodes and |E| = m edges. We investigate a linear size MIP formulation, referred to as (MIP-MaxCut), which can easily be derived via a standard linearization technique. However, the efficiency of the Branchand-Bound procedure applied to this formulation does not seem to have been investigated so far in the literature. Branch-and-bound based approaches for Max-Cut usually use the semi-metric polytope which has either an exponential size formulation consisting of the cycle inequalities or a compact size formulation consisting of O(mn) triangle inequalities [2], [16]. However, optimizing over the semi-metric polytope can be computationally demanding due to the slow convergence of cutting-plane algorithms and the high degeneracy of formulations based on the triangle inequalities. In this paper, we exhibit new structural properties of (MIP-MaxCut) that link the binary variables with the cycle inequalities. In particular, we show that fixing a binary variable at 0 or 1 in (MIP-MaxCut) can result in imposing the integrity of several original variables and the satisfaction of a possibly exponential number of cycle inequalities in the semi-metric formulation. Numerical results show that for sparse instances of Max-Cut, our approach exploiting this capability outperforms the branch-and-cut algorithms based on semi-metric polytope when implemented on the same framework; and even without any extra sophistication, the approach is capable of solving hard instances of Max-Cut within acceptable CPU times. The work is supported by the Programme Gaspard Monge pour Optimisation (PGMO).
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Submitted on : Sunday, November 22, 2020 - 9:51:58 AM
Last modification on : Tuesday, January 4, 2022 - 5:51:17 AM
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Viet Hung Nguyen, Michel Minoux. Linear size MIP formulation of Max-Cut: new properties, links with cycle inequalities and computational results. Optimization Letters, Springer Verlag, 2020, ⟨10.1007/s11590-020-01667-z⟩. ⟨hal-03018187⟩



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