Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

Abstract

We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a "nearly optimal" solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process.We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem.The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. The Max-Cut ProblemThe Max-Cut problem is one of the basic NP-hard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g.,

Semidefinite relaxations for non-convex quadratic mixed-integer programming

“…A well-known and very successful method for solving such problems relies on a strong dual bound given by an SDP relaxation of the problem. This relaxation results from an exact problem formulation by dropping a (non-convex) rank-one constraint [10]; combining this approach with additional cutting planes yields a very fast algorithm for solving the maximum cut problem [15].…”

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

“…For example, the quadratic assignment problem (QAP) is a BQP with problem structure based on multiplying flow and distance matrices (Anstreicher, 2003;Loiola et al, 2007). The max-cut problem, which maximizes the weights on the edges in an undirected graph (Rendl et al, 2010), and the maximum clique problem (Bomze et al, 1999) are other classic problems that may be formulated as MIQP with special mathematical structure enabling solution of large-scale instances. One common way of representing MIQCQP is via an undirected graph representation; see in Figure 1 some of the special structure patterns formed by MIQCQP including process networks, computational geometry, and MIQP .…”

Faster, but weaker, relaxations for quadratically constrained quadratic programs

“…We collected a total of 400 instances of (1) from the literature, which consisted of three groups: (i) 199 instances of the maximum cut (MaxCut) problem coming from [8] and [16] (21 instances of the Gset library and 178 instances of the BiqMac library, respectively); (ii) 64 instances of binary quadratic programming (BinQP) coming from the BiqMac library [16]; and (iii) 36 instances from GlobalLib [5] having bounded feasible sets. In particular, all instances had between n = 16 and n = 800 variables.…”