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 Branch-and-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 (Barahona and Mahjoub in Math Prog 36:157–173, 1986; Nguyen and Minoux in Networks 69(1):142–150, 2017). 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.

我们考虑具有\（| V | = n \）节点和\（| E | = m \）边的无向图\（G =（V，E）\）上的Max-Cut问题。我们研究了线性尺寸MIP公式，称为（MIP-MaxCut），可以通过标准线性化技术轻松得出。然而，迄今为止，尚未在文献中研究应用于该制剂的分支定界法的效率。用于Max-Cut的基于分支定界的方法通常使用半度量多面体，其具有由循环不等式组成的指数大小公式或由O（mn）三角形不等式（Barahona and Mahjoub in Math Prog 36：157–173，1986; Nguyen and Minoux in Networks 69（1）：142–150，2017）。然而，由于切平面算法的缓慢收敛以及基于三角形不等式的配方的高简并性，在半度量多面体上进行优化可能需要进行计算。在本文中，我们展示了（MIP-MaxCut）的新结构特性，该特性将二进制变量与循环不等式联系在一起。特别是，我们表明将二进制变量固定为（MIP-MaxCut）中的0或1可以导致强加几个原始变量的完整性，并满足半度量公式中可能指数级的循环不等式的要求。数值结果表明，对于Max-Cut稀疏实例，当在同一框架上实施时，我们利用此功能的方法要优于基于半度量多面体的分支剪切算法。即使没有任何复杂性，该方法也能够在可接受的CPU时间内解决Max-Cut的硬实例。