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Combining Reinforcement Learning and Constraint Programming for Combinatorial Optimization (2006.01610v1)

Published 2 Jun 2020 in cs.AI and cs.LG

Abstract: Combinatorial optimization has found applications in numerous fields, from aerospace to transportation planning and economics. The goal is to find an optimal solution among a finite set of possibilities. The well-known challenge one faces with combinatorial optimization is the state-space explosion problem: the number of possibilities grows exponentially with the problem size, which makes solving intractable for large problems. In the last years, deep reinforcement learning (DRL) has shown its promise for designing good heuristics dedicated to solve NP-hard combinatorial optimization problems. However, current approaches have two shortcomings: (1) they mainly focus on the standard travelling salesman problem and they cannot be easily extended to other problems, and (2) they only provide an approximate solution with no systematic ways to improve it or to prove optimality. In another context, constraint programming (CP) is a generic tool to solve combinatorial optimization problems. Based on a complete search procedure, it will always find the optimal solution if we allow an execution time large enough. A critical design choice, that makes CP non-trivial to use in practice, is the branching decision, directing how the search space is explored. In this work, we propose a general and hybrid approach, based on DRL and CP, for solving combinatorial optimization problems. The core of our approach is based on a dynamic programming formulation, that acts as a bridge between both techniques. We experimentally show that our solver is efficient to solve two challenging problems: the traveling salesman problem with time windows, and the 4-moments portfolio optimization problem. Results obtained show that the framework introduced outperforms the stand-alone RL and CP solutions, while being competitive with industrial solvers.

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Authors (5)
  1. Quentin Cappart (25 papers)
  2. Thierry Moisan (1 paper)
  3. Louis-Martin Rousseau (18 papers)
  4. Isabeau Prémont-Schwarz (10 papers)
  5. Andre Cire (3 papers)
Citations (126)
View on Semantic Scholar

Summary

Combining Reinforcement Learning and Constraint Programming for Combinatorial Optimization

The paper presents a novel approach to tackling combinatorial optimization problems (COPs) by integrating deep reinforcement learning (DRL) and constraint programming (CP). The authors address two significant limitations of existing DRL methods applied to NP-hard combinatorial tasks: their focus on the travelling salesman problem (TSP) and their lack of a systematic method for proving or improving approximate solutions. The integration of DRL with CP offers a pathway to generate high-quality solutions while providing guarantees of optimality.

Overview of Approach

The proposed method leverages dynamic programming (DP) principles to bridge the DRL and CP techniques. By encapsulating the DP model within both a reinforcement learning environment and a constraint programming model, the paper provides a unified framework that draws on the strengths of both strategies. This integration allows the solver to maintain optimal solution guarantees, unlike standard heuristics, while utilizing learned heuristics to guide the search process effectively.

Key Contributions

  1. Encoding Mechanism: The design of an encoding that converts a DP model of a COP into both a reinforcement learning environment and a CP model.
  2. Training Procedures: Utilization of deep Q-learning (DQN) and proximal policy optimization (PPO) to train RL models, which are then used to inform CP decisions like branching strategies.
  3. Search Strategies: The introduction of learned heuristics into CP search strategies, including branch-and-bound, iterative limited discrepancy search, and restart-based search.
  4. Empirical Validation: Demonstrated efficacy on solving the traveling salesman problem with time windows (TSPTW) and the 4-moments portfolio optimization problem (PORT), with superior results over isolated RL or CP solutions.
  5. Open Source Code: Commitment to fostering future research by making the source code and models openly available.

Experimental Results

Experimental results indicate that the hybrid approach outperforms both standalone CP and traditional RL strategies on the studied problems. In the case of the TSPTW, the hybrid methods provided more robust performance across increasing problem sizes, achieving high success rates in finding solutions and proving optimality compared to isolated methods. The PORT results further demonstrated that the approach can address non-linear objective functions effectively even in scenarios where conventional solvers struggle to prove optimality due to non-convexity or discrete coefficients.

Implications for AI and Optimization

The integration of DRL with CP as outlined in this paper suggests a promising direction for combinatorial optimization, particularly in solving NP-hard problems where traditional methods alone may fall short. By ensuring both solution quality and optimality proofs, this technique presents a substantial advancement for practical applications across diverse fields from logistics to finance.

Future Prospects

Looking ahead, the approach outlined in the paper may serve as a foundation for new hybrid models, particularly in leveraging machine learning advancements to enhance optimization frameworks. One area for future development could involve scaling the current methodology to handle bigger datasets and more complex problems. As computational resources and algorithms continue to advance, the scope for integrating learning models with optimization strategies will likely expand, presenting new opportunities for efficiency improvements and practical applications.

In conclusion, this paper provides a robust framework for integrating deep learning with optimization techniques in solving complex combinatorial problems, suggesting significant implications for advancing computational efficacy and reliability in decision-making processes.

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