Solving the Max-Cut Problem using Semidefinite Optimization in a Cutting Plane Algorithm.

Abstract

A central graph theory problem that occurs in experimental physics, circuit layout, and computational linear algebra is the max-cut problem. The max-cut problem is to find a bipartition of the vertex set of a graph with the objective to maximize the number of edges between the two partitions. The problem is NP-hard, i.e., there is no efficient algorithm to solve the max-cut problem to optimality.

We propose a semidefinite programming based cutting plane algorithm to solve the max-cut problem to optimality in this thesis. Semidefinite programming (SDP) is a convex optimization problem, where the variables are symmetric matrices. An SDP has a linear objective function, linear constraints, and also convex constraints requiring the matrices to be positive semidefinite. Interior point methods can efficiently solve SDPs and several software implementations like SDPT-3 are currently available.

Each iteration of our cutting plane algorithm has the following features: (a) an SDP relaxation of the max-cut problem, whose objective function provides an upper bound on the max-cut value, (b) the Goemans-Williamson heuristic to round the solution to the SDP relaxation into a feasible cut vector, that provides a lower bound on the max-cut value, and (c) a separation oracle that returns cutting planes to cut off the optimal solution to the SDP relaxation that is not in the max-cut polytope. Steps (a), (b), and (c) are repeated until the algorithm finds an optimal solution to the max-cut problem.

We have implemented the above cutting plane algorithm in MATLAB. Step (a) of the program uses SDPT-3 a primal-dual interior point software for solving the SDP relaxations. Step (c) of the algorithm returns triangle inequalities specific to the max-cut problem as cutting planes. We report our computational results with the algorithm on randomly generated graphs, where the number of vertices and the density of the edges vary between 5 to 50 and 0.1 to 1.0, respectively.