Partitioning through projections: Strong SDP bounds for large graph partition problems

The graph partition problem (GPP) aims at clustering the vertex set of a into fixed number disjoint subsets given sizes such that sum weights edges joining different sets is minimized. This paper investigates quality doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables are both positive semidefinite and nonnegative, strengthened by additional polyhedral cuts for two variations GPP: k-equipartition bisection problem. After reducing size facial reduction, we solve them cutting-plane algorithm combines an augmented Lagrangian method with Dykstra’s projection algorithm. Since many components our general, suitable solving various DNN large cutting planes. We first to show power planes GPP on benchmark instances up 1,024 vertices. Computational results impressive improvements in bounds.