Tackling Prevalent Conditions in Unsupervised Combinatorial Optimization: Cardinality, Minimum, Covering, and More
Fanchen Bu, Hyeonsoo Jo, Soo Yong Lee, Sungsoo Ahn, Kijung Shin
TL;DR
This work advances unsupervised learning for combinatorial optimization by targeting prevalent CO conditions (cardinality, minimum, covering, and uncertainty) through principled probabilistic objectives and fast greedy incremental derandomization. It introduces UCom2, a framework that enforces theoretical targets (differentiable, tight upper-bounded, and retaining the original objective minimum) and provides concrete objective and derandomization constructions for each condition, including extensions to non-binary decisions. The authors instantiate these derivations for facility location, maximum coverage, and robust coloring, and demonstrate superior speed-quality trade-offs on synthetic and real-world graphs, with thorough ablations validating the contributions of good objectives, greedy derandomization, and incremental updates. By offering a modular, theoretically grounded approach, this work enables more reliable differentiable optimization for CO under common constraints, with practical impact for scalable graph-based decision making.
Abstract
Combinatorial optimization (CO) is naturally discrete, making machine learning based on differentiable optimization inapplicable. Karalias & Loukas (2020) adapted the probabilistic method to incorporate CO into differentiable optimization. Their work ignited the research on unsupervised learning for CO, composed of two main components: probabilistic objectives and derandomization. However, each component confronts unique challenges. First, deriving objectives under various conditions (e.g., cardinality constraints and minimum) is nontrivial. Second, the derandomization process is underexplored, and the existing derandomization methods are either random sampling or naive rounding. In this work, we aim to tackle prevalent (i.e., commonly involved) conditions in unsupervised CO. First, we concretize the targets for objective construction and derandomization with theoretical justification. Then, for various conditions commonly involved in different CO problems, we derive nontrivial objectives and derandomization to meet the targets. Finally, we apply the derivations to various CO problems. Via extensive experiments on synthetic and real-world graphs, we validate the correctness of our derivations and show our empirical superiority w.r.t. both optimization quality and speed.