An exact algorithm for graph partitioning

• Published in: Mathematical programming Vol. 137; no. 1-2; pp. 531 - 556
• Main Authors: Hager, William W., Phan, Dzung T., Zhang, Hongchao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2013; Springer Nature B.V
• Subjects: Algorithms; Analysis; Branch & bound algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Eigenvalues; Full Length Paper; Graph theory; Graphs; Heuristic; Linear programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Partitioning; Programming; Quadratic programming; Semidefinite programming; Studies; Theoretical; Graph partitioning; 90C20; Affine underestimate; 90C35; 90C46; Min-cut; Branchand bound; 90C27; Quadratic programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0503-x

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.