DualOpt: A Dual Divide-and-Optimize Algorithm for the Large-scale Traveling Salesman Problem

TL;DR

DualOpt tackles the large-scale TSP by integrating a grid-based divide-and-conquer stage with an edge-breaking reduction, followed by a path-based optimization stage that employs a neural solver for sub-paths. The grid phase uses $K = 2^{N_iter - iter} \times 2^{N_iter - iter}$ grids and solves subproblems with LKH3 before merging into a reduced instance, while the path phase refines the tour by optimizing sub-paths via a batch neural solver under a REINFORCE objective. Empirical results on random TSP instances up to $N = 100{,}000$ and TSPLIB show DualOpt achieving competitive gaps (up to $1.40\%$) and substantial speed-ups (up to $104\times$) over LKH3, with strong generalization across distributions. The work demonstrates that a hybrid divide-and-conquer framework combined with learning-based refinement can effectively scale to real-world TSP applications and potentially extend to other combinatorial optimization problems.

Abstract

This paper proposes a dual divide-and-optimize algorithm (DualOpt) for solving the large-scale traveling salesman problem (TSP). DualOpt combines two complementary strategies to improve both solution quality and computational efficiency. The first strategy is a grid-based divide-and-conquer procedure that partitions the TSP into smaller sub-problems, solving them in parallel and iteratively refining the solution by merging nodes and partial routes. The process continues until only one grid remains, yielding a high-quality initial solution. The second strategy involves a path-based divide-and-optimize procedure that further optimizes the solution by dividing it into sub-paths, optimizing each using a neural solver, and merging them back to progressively improve the overall solution. Extensive experiments conducted on two groups of TSP benchmark instances, including randomly generated instances with up to 100,000 nodes and real-world datasets from TSPLIB, demonstrate the effectiveness of DualOpt. The proposed DualOpt achieves highly competitive results compared to 10 state-of-the-art algorithms in the literature. In particular, DualOpt achieves an improvement gap up to 1.40% for the largest instance TSP100K with a remarkable 104x speed-up over the leading heuristic solver LKH3. Additionally, DualOpt demonstrates strong generalization on TSPLIB benchmarks, confirming its capability to tackle diverse real-world TSP applications.

Figures (1)

• Figure 1: An illustration of the Grid-based Divide-and-Conquer Procedure with $N_{iter} = 3$.