An exact algorithm for graph partitioning

TL;DR

This paper introduces an exact branch and bound algorithm for solving edge weighted graph partitioning problems, leveraging convex-concave decomposition and semidefinite programming to improve bounds.

Contribution

It presents a novel continuous quadratic programming formulation and an efficient bounding technique using affine underestimates based on sphere centers.

Findings

Algorithm is competitive with state-of-the-art methods

Effective bounds improve solution efficiency

Demonstrates practical applicability to graph partitioning

Abstract

An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function Hessian, or by a diagonal shift obtained by solving a semidefinite programming problem. Numerical results show that the proposed algorithm is competitive with state-of-the-art graph partitioning codes.