ADMM for combinatorial graph problems

TL;DR

This paper develops ADMM-based algorithms for solving complex combinatorial graph problems like MAX-CUT and community detection, proving convergence to stationary points and demonstrating scalability through various benchmarks.

Contribution

It introduces two novel reformulations of combinatorial graph problems for ADMM, proving convergence despite nonconvex constraints and showing practical effectiveness on large-scale problems.

Findings

ADMM converges to stationary points in nonconvex graph problems.

The reformulations enable scalable solutions for MAX-CUT and community detection.

Experimental results demonstrate effectiveness on benchmark datasets.

Abstract

We investigate a class of general combinatorial graph problems, including MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints and solved via the alternating direction method of multipliers (ADMM). We propose two reformulations: one using vector variables and a binary constraint, and the other further reformulating the Burer-Monteiro form for simpler subproblems. Despite the nonconvex constraint, we prove the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this latter form, when the matrix factors are wide enough, local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation benchmark and simulated examples.