Geometric Algorithms for Neural Combinatorial Optimization with Constraints

TL;DR

This paper introduces a novel geometric framework for neural combinatorial optimization with constraints, enabling self-supervised learning and efficient feasible solution rounding for various constrained problems.

Contribution

It presents an end-to-end differentiable method leveraging convex geometry to handle discrete constraints in neural combinatorial optimization.

Findings

Outperforms neural baselines in cardinality-constrained optimization

Enables application to diverse combinatorial problems like graph independent sets and matroids

Provides a geometric decomposition approach for feasible solution generation

Abstract

Self-Supervised Learning (SSL) for Combinatorial Optimization (CO) is an emerging paradigm for solving combinatorial problems using neural networks. In this paper, we address a central challenge of SSL for CO: solving problems with discrete constraints. We design an end-to-end differentiable framework that enables us to solve discrete constrained optimization problems with neural networks. Concretely, we leverage algorithmic techniques from the literature on convex geometry and Carath\'eodory's theorem to decompose neural network outputs into convex combinations of polytope corners that correspond to feasible sets. This decomposition-based approach enables self-supervised training but also ensures efficient quality-preserving rounding of the neural net output into feasible solutions. Extensive experiments in cardinality-constrained optimization show that our approach can consistently outperform neural baselines. We further provide worked-out examples of how our method can be applied beyond cardinality-constrained problems to a diverse set of combinatorial optimization tasks, including finding independent sets in graphs, and solving matroid-constrained problems.