# Improving Optimization Bounds using Machine Learning: Decision Diagrams meet Deep Reinforcement Learning

arXiv: https://arxiv.org/abs/1809.03359 · HTML: https://mlsanity.com/paper/1809.03359/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2018.

## Built on by (most cited first, 13 of 13)

- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [Deep Reinforcement Learning for Exact Combinatorial Optimization: Learning to Branch](https://mlsanity.com/paper/2206.06965/) (2022)
- [Are Graph Neural Networks Optimal Approximation Algorithms?](https://mlsanity.com/paper/2310.00526/) (2023)
- [DOGE-Train: Discrete Optimization on GPU with End-to-end Training](https://mlsanity.com/paper/2205.11638/) (2022)
- [Vulcan: Solving the Steiner Tree Problem with Graph Neural Networks and Deep Reinforcement Learning](https://mlsanity.com/paper/2111.10810/) (2021)
- [Understanding Curriculum Learning in Policy Optimization for Online Combinatorial Optimization](https://mlsanity.com/paper/2202.05423/) (2022)
- [Deployment Optimization for Shared e-Mobility Systems with Multi-agent Deep Neural Search](https://mlsanity.com/paper/2111.02149/) (2021)
- [LEO: Learning Efficient Orderings for Multiobjective Binary Decision Diagrams](https://mlsanity.com/paper/2307.03171/) (2023)
- [Finding Influencers in Complex Networks: An Effective Deep Reinforcement Learning Approach](https://mlsanity.com/paper/2309.07153/) (2023)
- [Heuristic Multiobjective Discrete Optimization using Restricted Decision Diagrams](https://mlsanity.com/paper/2403.02482/) (2024)
- [DOPPLER: Dual-Policy Learning for Device Assignment in Asynchronous Dataflow Graphs](https://mlsanity.com/paper/2505.23131/) (2025)
- [A Clustering-Based Variable Ordering Framework for Relaxed Decision Diagrams for Maximum Weighted Independent Set Problem](https://mlsanity.com/paper/2512.15198/) (2025)
- [Reinforcement Learning in Operational Research: A Technical Review and Practical Roadmap](https://mlsanity.com/paper/2609.24750/) (2026)

## Abstract (arXiv)

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower bounds that can be significantly better than classical bounding mechanisms, such as linear relaxations. It is well known that the quality of the bounds achieved through this flexible bounding method is highly reliant on the ordering of variables chosen for building the diagram, and finding an ordering that optimizes standard metrics is an NP-hard problem. In this paper, we propose an innovative and generic approach based on deep reinforcement learning for obtaining an ordering for tightening the bounds obtained with relaxed and restricted DDs. We apply the approach to both the Maximum Independent Set Problem and the Maximum Cut Problem. Experimental results on synthetic instances show that the deep reinforcement learning approach, by achieving tighter objective function bounds, generally outperforms ordering methods commonly used in the literature when the distribution of instances is known. To the best knowledge of the authors, this is the first paper to apply machine learning to directly improve relaxation bounds obtained by general-purpose bounding mechanisms for combinatorial optimization problems.