Routing problems are a class of combinatorial problems with many practical applications. Recently, end-to-end deep learning methods have been proposed to learn approximate solution heuristics for such problems. In contrast, classical dynamic programming (DP) algorithms guarantee optimal solutions, but scale badly with the problem size. We propose Deep Policy Dynamic Programming (DPDP), which aims to combine the strengths of learned neural heuristics with those of DP algorithms. DPDP prioritizes and restricts the DP state space using a policy derived from a deep neural network, which is trained to predict edges from example solutions. We evaluate our framework on the travelling salesman problem (TSP), the vehicle routing problem (VRP) and TSP with time windows (TSPTW) and show that the neural policy improves the performance of (restricted) DP algorithms, making them competitive to strong alternatives such as LKH, while also outperforming most other 'neural approaches' for solving TSPs, VRPs and TSPTWs with 100 nodes.

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Most solution methods for solving large vehicle routing and scheduling problems are based on local search. A drawback of these approaches is that they are designed and optimized for specific types of vehicle routing problems (VRPs). As a consequence, it is hard to adapt these solution methods to handle new restrictions, without losing solution quality. We present a new framework for solving VRPs that can handle a wide range of different types of VRPs. Within this framework, restricted dynamic programming is applied to the VRP through the giant-tour representation. This algorithm is a construction heuristic which finds provably optimal solutions when unrestricted. We demonstrate the flexibility of the framework for a wide variety of different types of VRPs. The quality of solutions found by the framework is demonstrated by solving a set of benchmark instances for the capacitated VRP. The computational experiments show that restricted dynamic programming, which is a construction heuristic, develops routes of high quality. Therefore, this new framework for solving VRPs is highly valuable in practice.