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

TL;DR

This paper introduces a scalable Bregman-Sinkhorn algorithm for approximating the maximum-weight independent set problem, leveraging smoothed LP relaxations and a novel duality gap estimation to efficiently find high-quality solutions for large graphs.

Contribution

It presents a new dual coordinate descent method with an innovative smoothing schedule and a projection technique, improving solution quality and speed for large-scale NP-hard problems.

Findings

Outperforms standard smoothing techniques in efficiency.

Effectively solves large graphs with hundreds of thousands of nodes.

Produces high-quality approximate solutions within seconds.

Abstract

We propose a scalable approximate algorithm for the NP-hard maximum-weight independent set problem. The core component of our algorithm is a dual coordinate descent applied to a smoothed LP relaxation of the problem. This technique is commonly known by the names Bregman method and Sinkhorn algorithm in the literature. Our algorithm addresses a family of clique cover LP relaxations, where the constraints are determined by the set of cliques covering the underlying graph. The objective function of the relaxation is smoothed with an entropy term. A crucial aspect determining efficiency of our approach is controlling the smoothing level during the optimization process. While several dedicated techniques have been considered in the literature to this end, we propose a new one based on estimation of the relaxed duality gap. To make this estimation possible, we developed a new projection method to the feasible set of the considered LP relaxation. We experimentally show that our smoothing scheduling significantly outperforms the standard one based on feasibility estimation. Additionally to solving the relaxed dual, we utilize a simple and very efficient primal heuristic to obtain feasible integer solutions of the original non-relaxed problem. Our heuristic is a combination of randomized greedy generation and optimal recombination applied to the reduced costs computed by the dual optimization. Our experimental validation considers two datasets rooted in real-world applications, where our method demonstrates the ability to discover high-quality approximate solutions within 10 seconds for graphs with up to 882 thousand nodes and 344 million edges.