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Utku Umur Acikalin

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AAAI Conference 2026 Conference Paper

Unsupervised Combinatorial Probabilistic Reasoning: Probabilistic Coin Change Problem

Zhongdi Qu
Yingheng Wang
Utku Umur Acikalin
Aaron M. Ferber
Goncalo J. Gouveia
Brandon Bills
Guohui Li
Joshua Kline

We introduce the Probabilistic Coin Change Problem (PCCP), a novel variant of the classical Combination Coin Change Problem (CCCP), motivated by a real-world scientific inverse task. The goal of CCCP is to enumerate all unordered combinations of coin denominations that sum to a given target. In PCCP, each coin type’s value follows a discrete probability distribution, and the aggregate value of a combination of coins is thus stochastic. Given a set of such coin types and noisy observations of total sums, the task is to infer the most likely latent coin combination. To address the combinatorial and probabilistic complexity of PCCP, we propose DeepProReasoner (Deep Combinatorial Probabilistic Reasoning with Embedded Representations), an unsupervised, end-to-end, deep-learning framework that integrates combinatorial reasoning, latent-space modeling, and differentiable probabilistic reasoning. The model is trained using a reconstruction loss between the observed empirical distribution and a decoded probability mass function (PMF), enabling efficient gradient-based search over a continuous relaxation of the combinatorial space. We evaluate DeepProReasoner on two instances of PCCP: (1) a synthetic Candy Mix problem for ablation studies, and (2) a real-world task of molecular formula inference from ultrahigh resolution mass spectrometry (MS) data. Besides the two given instances, PCCP captures a wide range of inverse settings in biology, chemistry, environmental sciences, and medicine, where latent combinatorial structures give rise to noisy aggregate observations through stochastic processes. Our results show that DeepProReasoner achieves high accuracy and robustness, outperforming state-of-the-art methods.

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ICLR Conference 2025 Conference Paper

Learning to Explore and Exploit with GNNs for Unsupervised Combinatorial Optimization

Utku Umur Acikalin
Aaron M. Ferber
Carla P. Gomes

Combinatorial optimization (CO) problems are pervasive across various domains, but their NP-hard nature often necessitates problem-specific heuristic algorithms. Recent advancements in deep learning have led to the development of learning-based heuristics, yet these approaches often struggle with limited search capabilities. We introduce Explore-and-Exploit GNN ($X^2$GNN, pronounced x-squared GNN), a novel unsupervised neural framework that combines exploration and exploitation for combinatorial search optimization: i) Exploration - $X^2$GNN generates multiple solutions simultaneously, promoting diversity in the search space; (ii) Exploitation - $X^2$GNN employs neural stochastic iterative refinement to exploit partial existing solutions, guiding the search toward promising regions and helping escape local optima. By balancing exploration and exploitation, $X^2$GNN achieves superior performance and generalization on several graph CO problems including Max Cut, Max Independent Set, and Max Clique. Notably, for large Max Clique problems, $X^2$GNN consistently generates solutions within 1.2\% of optimality, while other state-of-the-art learning-based approaches struggle to reach within 22\% of optimal. Moreover, $X^2$GNN consistently generates better solutions than Gurobi on large graphs for all three problems under reasonable time budgets. Furthermore, $X^2$GNN exhibits exceptional generalization capabilities. For the Maximum Independent Set problem, $X^2$GNN outperforms state-of-the-art methods even when trained on smaller or out-of-distribution graphs compared to the test set. Our framework offers a more effective and flexible approach to neural combinatorial optimization, addressing a key challenge in the field and providing a promising direction for future research in learning-based heuristics for combinatorial optimization.

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