Each AvehicleC starts sumofcityC, after again.

There are two more things to verify.

Notably, we offer fresh perspectives on how neural solvers can handle VRP constraints.

To this end, we introduce Poppy, a simple training procedure for populations.

We introduce Policy Optimization with Multiple Optima (POMO), anend-to-end approach forbuildingsuchaheuristic solver.POMO isapplicable to a wide range of CO problems.

We further evaluate SGBS+EAS on nine real-world based instance sets from [15]. Each instance set consists of 20 instances that have similar characteristics (i.e., they have been sampled from the same underlying distribution). To account for this new evaluation setting, we always perform 10 runs in parallel for EAS and SGBS+EAS. This improves the solution quality, while leading only to a slight increase of the requiredruntime. For SGBS+EAS we set (β, γ) = (35,5), the learning rate α = 0.005 and λ = 0.05.

Neural approaches for combinatorial optimization (CO) equip a learning mechanism to discover powerful heuristics for solving complex real-world problems. While neural approaches capable of high-quality solutions in a single shot are emerging, state-of-the-art approaches are often unable to take full advantage of the solving time available to them. In contrast, hand-crafted heuristics perform highly effective search well and exploit the computation time given to them, but contain heuristics that are difficult to adapt to a dataset being solved.

We present NeuroLKH, a novel algorithm that combines deep learning with the strong traditional heuristic Lin-Kernighan-Helsgaun (LKH) for solving Traveling Salesman Problem. Specifically, we train a Sparse Graph Network (SGN) with supervised learning for edge scores and unsupervised learning for node penalties, both of which are critical for improving the performance of LKH. Based on the output of SGN, NeuroLKH creates the edge candidate set and transforms edge distances to guide the searching process of LKH. Extensive experiments firmly demonstrate that, by training one model on a wide range of problem sizes, NeuroLKH significantly outperforms LKH and generalizes well to much larger sizes. Also, we show that NeuroLKH can be applied to other routing problems such as Capacitated Vehicle Routing Problem (CVRP), Pickup and Delivery Problem (PDP), and CVRP with Time Windows (CVRPTW).