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

arXiv: https://arxiv.org/abs/2405.08424 · HTML: https://mlsanity.com/paper/2405.08424/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 87 references that are in the map.
- First posted on arXiv 14 May 2024.

## Builds on (most cited first, 30 of 45)

- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [Reinforcement Learning for Combinatorial Optimization: A Survey](https://mlsanity.com/paper/2003.03600/) (2020)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [DIMES: A Differentiable Meta Solver for Combinatorial Optimization Problems](https://mlsanity.com/paper/2210.04123/) (2022)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs](https://mlsanity.com/paper/2006.10643/) (2020)
- [Combinatorial Optimization with Physics-Inspired Graph Neural Networks](https://mlsanity.com/paper/2107.01188/) (2021)
- [Learning Collaborative Policies to Solve NP-hard Routing Problems](https://mlsanity.com/paper/2110.13987/) (2021)
- [GLOP: Learning Global Partition and Local Construction for Solving Large-scale Routing Problems in Real-time](https://mlsanity.com/paper/2312.08224/) (2023)
- [Simulation-guided Beam Search for Neural Combinatorial Optimization](https://mlsanity.com/paper/2207.06190/) (2022)
- [DeepACO: Neural-enhanced Ant Systems for Combinatorial Optimization](https://mlsanity.com/paper/2309.14032/) (2023)
- [RL4CO: an Extensive Reinforcement Learning for Combinatorial Optimization Benchmark](https://mlsanity.com/paper/2306.17100/) (2023)
- [Empowering Graph Representation Learning with Test-Time Graph Transformation](https://mlsanity.com/paper/2210.03561/) (2022)
- [Reinforcement Learning with Combinatorial Actions: An Application to Vehicle Routing](https://mlsanity.com/paper/2010.12001/) (2020)
- [Winner Takes It All: Training Performant RL Populations for Combinatorial Optimization](https://mlsanity.com/paper/2210.03475/) (2022)
- [Unsupervised Learning for Solving the Travelling Salesman Problem](https://mlsanity.com/paper/2303.10538/) (2023)
- [Learning What to Defer for Maximum Independent Sets](https://mlsanity.com/paper/2006.09607/) (2020)
- [CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints](https://mlsanity.com/paper/2105.02343/) (2021)
- [Combinatorial Optimization with Policy Adaptation using Latent Space Search](https://mlsanity.com/paper/2311.13569/) (2023)
- [Let the Flows Tell: Solving Graph Combinatorial Optimization Problems with GFlowNets](https://mlsanity.com/paper/2305.17010/) (2023)
- [The SCIP Optimization Suite 9.0](https://mlsanity.com/paper/2402.17702/) (2024)
- [Unsupervised Learning for Combinatorial Optimization with Principled Objective Relaxation](https://mlsanity.com/paper/2207.05984/) (2022)

## Built on by (most cited first, 7 of 7)

- [Self-Supervised Transformers as Iterative Solution Improvers for Constraint Satisfaction](https://mlsanity.com/paper/2502.15794/) (2025)
- [Geometric Algorithms for Neural Combinatorial Optimization with Constraints](https://mlsanity.com/paper/2510.24039/) (2025)
- [FORGE: Foundational Optimization Representations from Graph Embeddings](https://mlsanity.com/paper/2508.20330/) (2025)
- [Preference-Based Gradient Estimation for ML-Guided Approximate Combinatorial Optimization](https://mlsanity.com/paper/2502.19377/) (2025)
- [On Training-Test (Mis)alignment in Unsupervised Combinatorial Optimization: Observation, Empirical Exploration, and Analysis](https://mlsanity.com/paper/2506.16732/) (2025)
- [Self-Supervised Combinatorial Optimization with Constraints via Frank-Wolfe](https://mlsanity.com/paper/2609.25728/) (2026)
- [Probabilistic Multi-Source Data Selection via Learnable Categorical Priors: Variance Reduction and Guaranteed Derandomization](https://mlsanity.com/paper/2610.14942/) (2026)

## Abstract (arXiv)

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.