A Bregman-Sinkhorn Algorithm for the Maximum Weight Independent Set Problem

Abstract

We propose a scalable approximate algorithm for the NP-hard maximum-weight independent set problem, based on dual coordinate descent applied to a smoothed clique-cover LP relaxation. Our method, a variant of the Bregman/Sinkhorn algorithm, employs entropy smoothing with a novel duality-gap-based smoothing scheduling strategy that empirically outperforms standard feasibility scheduling for the relaxed problem. Our new projection to the primal feasible set enables accurate duality gap estimation. Combined with a basic primal heuristic leveraging reduced costs, our approach yields high-quality integer solutions. On real-world datasets, it efficiently finds high-quality approximate solutions for graphs with up to 882,000 nodes and 344 million edges within seconds.