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).
Highlights
1.1 The Max-Cut problemLet G = (V, E) be an undirected graph with n = |V | and m = |E|
– As an experimental confirmation of our analysis, we present and discuss in Section 4 a series of computational results on Max-Cut for a set of large size sparse graph instances obtained by applying a standard Branch-and-Bound solver to (MIP-MaxCut) without using any of the sophisticated techniques previously proposed in the existing literature for the exact solution of such large scale problems
Since the purpose of the present lemma is both to prove Theorem 1 and to provide a constructive way of strengthening the new linear size formulation to be discussed in Section 3, we provide the full proof in the appendix for self-containedness. ⇐ Let x ∈ [0, 1]E be a vector satisfying all the cycle inequalities associated with C
Summary
G could be the complete graph Kn = (Vn, En) of n nodes. We denote by ij the edge between the two nodes i and j of V. A cut in G associated with a node subset S ⊂ V , denoted δ(S), is the set of the edges that have exactly one end-node in S. The. Max-Cut problem is to find a cut of maximum total weight or equivalently to find a node subset S such that ij∈δ(S) cij is maximum. For each cut δ(S), the incidence vector associated with δ(S) is a vector χ(δ(S)) ∈ {0, 1}E where χ(δ(S))ij =. Finding a maximum weight cut is equivalent to optimizing over the cut polytope CUTP(G) which is the convex hull of the incidence vectors associated with the cuts in G
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