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

TL;DR

This paper demonstrates that strengthened doubly nonnegative relaxations with cutting planes provide highly effective bounds for large graph partition problems, significantly improving solution quality on large benchmarks.

Contribution

It introduces a novel approach combining DNN relaxations, facial reduction, and cutting-plane algorithms to solve large graph partition problems more effectively.

Findings

DNN relaxations with cuts outperform previous bounds

Effective for graphs up to 1024 vertices

Significant computational improvements achieved

Abstract

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