Semidefinite programming for assignment and partitioning problems

Abstract

Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for N P-hard combinatorial optimization problems of simple structure such as the max-cut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems.

A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the so-called "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primal-dual interior-point algorithm by using a conjugate gradient method and by taking advantage of the special data structure of our relaxation. Numerical tests show that the approximations given by our approach are of high quality.

Future work for solving a large sparse problem with our approach is also discussed for each of the applications. In particular, for a large sparse set partitioning problem, we propose an approach combining a mixed LP-SDP relaxation with matrix decomposition techniques.