Dataless Quadratic Neural Networks for the Maximum Independent Set Problem

2024-06-27

arXiv.org

AI Summary

This paper addresses the NP-hard Maximum Independent Set (MIS) problem on graphs by proposing a data-agnostic, continuously differentiable quadratic neural optimization framework. Methodologically, it first reformulates MIS as a Maximum Clique problem via complement graph transformation and introduces a clique-informed semidefinite quadratic programming model, rigorously proving a one-to-one correspondence between MIS solutions and local minima of the model. Second, it designs a momentum-based gradient descent algorithm coupled with a theory-driven feasibility verification criterion—requiring neither labeled data nor distributional assumptions. Third, it employs multi-start parallel optimization and achieves linear-time inference complexity. Experiments on standard benchmarks demonstrate that the method significantly outperforms mainstream heuristics and sampling-based approaches in solution size and generalization capability, while inference time scales linearly with the number of vertices.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 for the MIS size, and characterize stationary points. To solve our non-convex objective, we propose solving parallel multiple initializations using momentum-based gradient descent, complemented by an efficient MIS checking criterion derived from our theory. Therefore, 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.

Problem

• Quadratic Neural Networks
• Maximum Independent Set
• Combinatorial Optimization

Innovation

• Quadratic Neural Networks
• Parallel Momentum Gradient Descent
• Clique-Informed Optimization