Tackling Prevalent Conditions in Unsupervised Combinatorial Optimization: Cardinality, Minimum, Covering, and More

TL;DR

This paper advances unsupervised combinatorial optimization by developing theoretically justified objectives and derandomization methods for prevalent conditions, demonstrating improved performance on synthetic and real-world graphs.

Contribution

It introduces novel, theoretically grounded objectives and derandomization techniques tailored for common conditions in unsupervised CO, filling key gaps in existing methods.

Findings

Validated the correctness of derived objectives and derandomization methods.

Achieved superior optimization quality compared to existing approaches.

Demonstrated faster convergence and better results on diverse graph datasets.

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.