Parallel Semidefinite Programming and Combinatorial Optimization

Abstract

The use of semidefinite programming in combinatorial optimization continues to grow. This growth can be attributed to at least three factors: new semidefinite relaxations that provide tractable bounds to hard combinatorial problems, algorithmic advances in the solution of semidefinite programs (SDP), and the emergence of parallel computing. Solution techniques for minimizing combinatorial problems often involve approximating the convex hull of the solution set and establishing lower bounds on the solution. Polyhedral approximations that use linear programming (LP) methodologies have been successful for many combinatorial problems, but they have not been as successful for problems such as maximum cut and maximum stable set. The semidefinite approximation for the stable set problem was proposed by Grotschel, Lovasz, and Shrijver [32] and further developed by Alizadeh [1], Polijak, Rendl, and Wolkowicz [48], and many other authors. The Lovasz number [43] is the solution of an SDP that provides an upper bound to the maximum clique of a graph and a lower bound to its chromatic number. Tractable bounds have also been provided for MAXCUT [30], graph bisection [47], MAX k-CUT [27, 24], graph coloring [33], and the quadratic assignment problem [61]. Many more combinatorial problems can be relaxed into a semidefinite program, and some of these relaxations offer a projection back to a feasible solution that is guaranteed to be within a specified fraction of optimality. These combinatorial problems are NP hard in the general case, so even approximate solutions are difficult to find. Most polyhedral relaxations do not offer a performance guarantee, but Geomans and