Research Reviews of Combinatorial Optimization Methods Based on Deep Reinforcement Learning
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摘要: 组合优化问题广泛存在于国防、交通、工业、生活等各个领域, 几十年来, 传统运筹优化方法是解决组合优化问题的主要手段, 但随着实际应用中问题规模的不断扩大、求解实时性的要求越来越高, 传统运筹优化算法面临着很大的计算压力, 很难实现组合优化问题的在线求解. 近年来随着深度学习技术的迅猛发展, 深度强化学习在围棋、机器人等领域的瞩目成果显示了其强大的学习能力与序贯决策能力. 鉴于此, 近年来涌现出了多个利用深度强化学习方法解决组合优化问题的新方法, 具有求解速度快、模型泛化能力强的优势, 为组合优化问题的求解提供了一种全新的思路. 因此本文总结回顾近些年利用深度强化学习方法解决组合优化问题的相关理论方法与应用研究, 对其基本原理、相关方法、应用研究进行总结和综述, 并指出未来该方向亟待解决的若干问题.Abstract: Combinatorial optimization problems widely exist in various fields such as national defense, transportation, industry and life. For decades, traditional operational research methods are the main means to solve combinatorial optimization problems. However, with the increase of problem size in practical applications and the increasing demands for real-time optimization, traditional methods suffer from great computational burdens, and it is difficult to realize the online solution of combinatorial optimization problems. In recent years, with the rapid development of deep learning technology, the achievements of deep reinforcement learning in AlphaGo, robot and other fields show its strong learning ability and sequential decision-making ability. In view of this, in recent years, a number of new methods using deep reinforcement learning to solve combinatorial optimization problems have emerged, which have the advantages of fast solving speed and strong model generalization ability. It provides a new idea for solving combinatorial optimization problems. Therefore, this paper summarizes and reviews the theoretical methods and application researches of this kind of method in recent years.
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表 1 现有算法模型、训练方法、求解问题、以及优化效果比较
Table 1 Comparison of model, training method, solving problems and performance with existing algorithms
方法类别 研究 模型以及训练方法 求解问题及优化效果 基于Pointer
Network的端
到端方法2015年Vinyals等人[30] Ptr-Net + 监督式训练 30 TSP问题: 接近最优解, 优于启发式算法. 40, 50-TSP:
与最优解存在一定差距. 凸包问题、三角剖分问题.2017年Bello等人[31] Ptr-Net + REINFORCE &
Critic baseline50-TSP: 优于[30]. 100-TSP: 接近Concorde最优解.
200-Knapsack: 达到最优解.2018年Nazari等人[32] Ptr-Net + REINFORCE &
Critic baseline100-TSP: 与[31]优化效果相近, 训练时间降低约60%.
100-CVRP/随机CVRP: 优于多个启发式算法.2018年Deudon等人[33] Transformer Attention +
REINFORCE & Critic baseline20, 50-TSP: 优于[31]. 100-TSP: 与[31]优化效果相近. 2019年Kool等人[34] Transformer Attention +
REINFORCE & Rollout baseline100-TSP: 优于[30-33,37,40]. 100-CVRP、100-SDVRP、100-OP、
100-PCTSP、SPCTSP: 接近Gurobi最优解,
优于多种启发式方法.2020年Ma等人[35] Graph Pointer Network + HRL 20, 50-TSP: 优于[31,37], 劣于[34]. 250, 500, 1000-TSP:
优于[31,34]. 20-TSPTW: 优于OR-Tools、蚁群算法.2020年Li等人[36] Ptr-Net + REINFORCE & Critic
baseline & 分解策略/参数迁移40, 100, 150, 200, 500-两目标/三目标TSP :
优于MOEA/D、NSGA-II、MOGLS.基于图神经网络
的端到端方法2017年Dai等人[37] structure2vec + DQN 1200-TSP: 接近[31]. 1200-MVC(最小顶点覆盖): 接近最优解.
1200-MAXCUT(最大割集): 接近最优解.2019年Mittal等人[38] GCN + DQN 2k至20k-MCP(最大覆盖问题): 优于[37]. 10k, 20k, 50k-MVC: 优于[37]. 2018年Li等人[39] GCN + 监督式训练 +
引导树搜索实际数据集MVC、MIS(最大独立点集)、MC(极大团)、
Satisfiability(适定性问题): 优于[37].2017年Nowak等人[40] GNN + 监督式训练 +
波束搜索20-TSP: 劣于[30]. 2019年Joshi等人[41] GCN + 监督式训练 +
波束搜索20, 50, 100-TSP: 略微优于[30,31,33,34], 优于[37]. 深度强化学习改
进的局部搜索
方法2019年Chen等人[47] Ptr-Net + Actor-Critic 20-CVRP: 达到最优解. 50, 100-CVRP: 优于[32,34]、OR-Tools.
作业车间调度: 优于OR-Tools、DeepRM2019年Yolcu等人[48] GNN + REINFORCE 实际数据集Satisfiability、MIS、MVC、MC、图着色问题: 更少
搜索步数得到最优解、但单步运行时间长于传统算法.2020年Gao等人[49] Graph Attention + PPO 100-CVPR: 优于[34]. 100-CVPRTW: 优于多个启发式方法.
400-CVRPTW: 劣于单个启发式方法, 优于其他.2020年Lu等人[50] Transformer Attention +
REINFORCE20, 50, 100-CVRP: 优于[32,34,47], 以及优于OR Tools、
LKH3. 且运行时间远低于LKH3.
下载: 导出CSV
表 2 端到端模型在TSP问题上优化性能比较.
Table 2 Comparison of end-to-end model on TSP.
方法类别 模型 TSP-20 TSP-50 TSP-100 最优 Concorde 3.84 5.70 7.76 基于指
针网络
(Attention
机制)Vinyals[30] 3.88 7.66 — Bello[31] 3.89 5.95 8.30 Nazari[32] 3.97 6.08 8.44 Deudon[33] 3.86 5.81 8.85 Deudon[33]+2OPT 3.85 5.85 8.17 Kool[34](greedy) 3.85(0s) 5.80(2s) 8.12(6s) Kool[34](sampling) 3.84(5m) 5.73(24m) 7.94(1h) 基于图神
经网络Dai[37] 3.89 5.99 8.31 Nowak[40] 3.93 — — Joshi[41](greedy) 3.86(6s) 5.87(55s) 8.41(6m) Joshi[41](BS) 3.84(12m) 5.70(18m) 7.87(40m)
下载: 导出CSV
表 4 不同组合优化问题求解算法统计与比较
Table 4 Summary and comparison of algorithms on different combinatorial optimization problems
组合优化问题 文献 模型细节 TSP问题 [30–36] 基于Ptr-Net架构
(Encoder-Decoder-Attention)[37] GNN+DQN [40,41] GNN+监督式训练+波束搜索 VRP问题 [32, 34] 基于Ptr-Net架构(Encoder-
Decoder-Attention)[47, 49, 50] DRL训练局部搜索算子. [47]:
Ptr-Net模型, [49]: Graph
Attention模型, [50]: Transformer
Attention模型.最小顶点覆盖问题(MVC) [37, 38, 48] GNN + RL [39] GNN + 监督式训练 + 树搜索 最大割集问题(MaxCut) [37] GNN + DQN [57] Message Passing Neural Network
(MPNN) + DQN[58] * CNN&RNN + PPO 适定性问题(Satisfiability) [39, 48] GNN + 监督式训练/RL 最小支配集问题 (MDS) [48] GNN + RL [59] * Decision Diagram + RL 极大团问题(MC) [39, 48] GNN + 监督式训练/RL 最大独立集问题(MIS) [39] GNN + 监督式训练 + 树搜索 [60] * GNN + RL + 蒙特卡洛树搜索 背包问题(Knapsack) [31] Ptr-Net + RL 车间作业调度问题 [47] LSTM + RL训练局部搜索算子 装箱问题(BPP) [61] * LSTM + RL [62] * NN + RL + 蒙特卡洛树搜索 图着色问题 [48] GNN + RL [63] * LSTM + RL + 蒙特卡洛树搜索
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