[Paper Review] NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem

Liang Xin, Wen Song+2 more|arXiv (Cornell University)|Oct 15, 2021

Vehicle Routing Optimization Methods34 references44 citations

TL;DR

NeuroLKH blends a Sparse Graph Network with the Lin-Kernighan-Helsgaun solver to learn edge scores and node penalties, guiding LKH for improved TSP solutions and extending to related routing problems.

ABSTRACT

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).

Motivation & Objective

• Motivate and address limitations of purely hand-crafted edge candidate sets and per-instance subgradient optimization in LKH.
• Develop a learning-based module (SGN) to produce edge scores and node penalties for TSP instances.
• Enable fast, generalized guidance to LKH by transforming edge distances and refining candidate edges.
• Demonstrate significant performance gains over LKH and VSR-LKH across sizes and distributions, and extend to CVRP, PDP, and CVRPTW.

Proposed method

• Represent TSP instances as sparse directed graphs and apply a Sparse Graph Network (SGN) to produce edge scores and node penalties.
• Train edge scores with supervised learning using optimal tour edges as targets.
• Train node penalties with unsupervised learning to steer Minimum 1-Tree degrees toward 2, akin to subgradient optimization.
• Transform edge distances via cij = sij + pi + pj using learned penalties, preserving the optimal tour but improving search guidance.
• Create an edge candidate set per node based on learned edge scores for guiding the lambda-opt search in LKH.
• Solve via multiple LKH trials, where SGN outputs guide distance transformation and candidate sets, reducing per-instance optimization time.

Experimental results

Research questions

• RQ1 Can a learned model produce edge scores that yield higher-quality edge candidate sets than traditional LKH-generated sets?
• RQ2 Can simultaneous learning of edge scores and node penalties generalize across a range of TSP instance sizes and distributions?
• RQ3 Does integrating SGN-guided distance transformation with LKH improve solution quality and runtime, especially for large-scale problems?
• RQ4 Can NeuroLKH extend effectively to related routing problems such as CVRP, PDP, and CVRPTW?
• RQ5 What is the impact of fast SGN inference versus traditional LKH subgradient optimization on overall performance?

Key findings

• NeuroLKH significantly outperforms LKH and VSR-LKH across training sizes (100, 200, 500 nodes) and time limits, reducing optimality gaps by large margins.
• A single SGN trained over a range of sizes generalizes to substantially larger problems (1000, 2000, 5000 nodes) with fast fine-tuning of node penalties.
• Inference time for SGN scales roughly linearly with problem size (3s to 208s over 100-5000 nodes), while LKH subgradient optimization time grows superlinearly, making NeuroLKH especially advantageous for short time limits.
• NeuroLKH generalizes well to TSPLIB Euclidean instances, solving hard instances more often than LKH in many cases, and shows strong results on larger distributions with a mixture-trained model (NeuroLKH_M).
• NeuroLKH can be extended to CVRP, PDP, and CVRPTW by learning edge candidate sets (node penalties do not apply to these problems).