Learning to Approximate Uniform Facility Location via Graph Neural Networks
Chendi Qian, Christopher Morris, Stefanie Jegelka, Christian Sohler
TL;DR
The paper tackles UniFL by blending algorithmic structure with differentiable learning through a fully unsupervised MPNN. It introduces a radius-informed, differentiable architecture that mirrors a simple distributed UniFL algorithm, yielding an $O( obreak ext{log}(n))$-approximation and provable size generalization to larger instances. Through a recursive extension, it achieves a constant-factor overall guarantee while maintaining efficiency. Empirically, the method outperforms classical approximation baselines, generalizes to much larger graphs without retraining, and remains competitive with ILP-based approaches, demonstrating a practical bridge between learning and approximation in discrete optimization.
Abstract
There has been a growing interest in using neural networks, especially message-passing neural networks (MPNNs), to solve hard combinatorial optimization problems heuristically. However, existing learning-based approaches for hard combinatorial optimization tasks often rely on supervised training data, reinforcement learning, or gradient estimators, leading to significant computational overhead, unstable training, or a lack of provable performance guarantees. In contrast, classical approximation algorithms offer such performance guarantees under worst-case inputs but are non-differentiable and unable to adaptively exploit structural regularities in natural input distributions. We address this dichotomy with the fundamental example of Uniform Facility Location (UniFL), a variant of the combinatorial facility location problem with applications in clustering, data summarization, logistics, and supply chain design. We develop a fully differentiable MPNN model that embeds approximation-algorithmic principles while avoiding the need for solver supervision or discrete relaxations. Our approach admits provable approximation and size generalization guarantees to much larger instances than seen during training. Empirically, we show that our approach outperforms standard non-learned approximation algorithms in terms of solution quality, closing the gap with computationally intensive integer linear programming approaches. Overall, this work provides a step toward bridging learning-based methods and approximation algorithms for discrete optimization.