Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

Abstract

We consider the maximum k-cut problem that involves partitioning the vertex set of a graph into k subsets such that the sum of the weights of the edges joining vertices in different subsets is maximized. The associated semidefinite programming (SDP) relaxation is known to provide strong bounds, but it has a high computational cost. We use a cutting plane algorithm that relies on the early termination of an interior point method, and we study the performance of SDP and linear programming (LP) relaxations for various values of k and instance types. The LP relaxation is strengthened using combinatorial facet-defining inequalities and SDP-based constraints. Our computational results suggest that the LP approach, especially with the addition of SDP-based constraints, outperforms the SDP relaxations for graphs with positive-weight edges and k ≥ 7.

Computational study of a branching algorithm for the maximum k-cut problem

“…An advantage of the SDP relaxation is that it provides strong bounds, especially because some hypermetric inequalities are implicit [12]. However, it has an expensive processing time: See the recent analysis of [3,45].…”

“…In [45], the authors proposed an SDP-based inequality to reinforce both edge-only or vertexand-edge relaxations of max-k-cut. The following constraint is based on the linear semi-infinite programming approach for generic SDPs [46]:…”

“…where the Euclidean norm of µ is typically one. Constraint (25) has an infinite number of rows, so a separation routine based on eigenvalues is also introduced in [45]. The computational results show that SDP-based inequalities are helpful when k ≥ 7.…”

“…For the LP relaxations, we investigate the performance of the SDP-based constraints since it has been shown in [45] that these constraints are strong but expensive. In this section we consider five methods:…”

“…CPA is an iterative method that solves a relaxation and then finds and adds to the relaxation some of the violated inequalities (or cutting planes). We apply the CPA proposed in [45] that implements the early-termination technique in an interior point method. In [45], at each iteration of the CPA, at most 2|V | of the most violated constraints are added and the resulting new problem is solved from scratch.…”