Differentiable Quadratic Optimization For The Maximum Independent Set Problem

TL;DR

This paper introduces pCQO-MIS, a differentiable quadratic optimization method that improves maximum independent set solutions by leveraging a novel formulation, multiple initializations, and theoretical insights, achieving better results efficiently.

Contribution

The paper proposes a new quadratic formulation for MIS incorporating a maximum clique term and a parallelized optimization approach with theoretical analysis, advancing differentiable combinatorial optimization.

Findings

pCQO-MIS achieves larger MIS sizes than existing methods.

The runtime scales with the number of nodes, not edges.

The method outperforms exact, heuristic, and data-centric approaches in experiments.

Abstract

Combinatorial Optimization (CO) addresses many important problems, including the challenging Maximum Independent Set (MIS) problem. Alongside exact and heuristic solvers, differentiable approaches have emerged, often using continuous relaxations of ReLU-based or quadratic objectives. Noting that an MIS in a graph is a Maximum Clique (MC) in its complement, we propose a new quadratic formulation for MIS by incorporating an MC term, improving convergence and exploration. We show that every maximal independent set corresponds to a local minimizer, derive conditions with respect to the MIS size, and characterize stationary points. To tackle the non-convexity of the objective, we propose optimizing several initializations in parallel using momentum-based gradient descent, complemented by an efficient MIS checking criterion derived from our theory. We dub our method as parallelized Clique-Informed Quadratic Optimization for MIS (pCQO-MIS). Our experimental results demonstrate the effectiveness of the proposed method compared to exact, heuristic, sampling, and data-centric approaches. Notably, our method avoids the out-of-distribution tuning and reliance on (un)labeled data required by data-centric methods, while achieving superior MIS sizes and competitive runtime relative to their inference time. Additionally, a key advantage of pCQO-MIS is that, unlike exact and heuristic solvers, the runtime scales only with the number of nodes in the graph, not the number of edges. Our code is available at the GitHub repository: https://github.com/ledenmat/pCQO-mis-benchmark/tree/refactor.