DualCert: A Solver for the Traveling Salesman Problem with Constraint-Coupled Learning

2026-08-09

AI Summary

This work addresses the challenge of solving large-scale Traveling Salesman Problems (TSP) under limited computational budgets by introducing a constraint-coupled learning mechanism that, for the first time, directly embeds degree constraints and dynamically separated subtour elimination constraints into the learning process. Built upon the KKT manifold, the method integrates constraint mirror descent with implicit differentiation for state updates and synergistically combines Held–Karp dual ascent, edge testing, and path construction to yield verifiable decisions within a fixed time budget. Experimental results on 1,000 TSP1000 instances demonstrate that the approach achieves solutions within 0.0573% of LKH-3’s quality on average, with a per-instance runtime of 9.55 seconds, an edge coverage rate of 81.46%, and a 67.1% reduction in average optimality gap compared to NeuroLKH.

Abstract

Large traveling salesman problem (TSP) instances require a solver to allocate limited computation while preserving the validity of its outputs. Existing neural--operations-research (OR) hybrids predict guidance without requiring learned transitions to satisfy constraints discovered during search. DualCert introduces \emph{constraint-coupled learning}, in which current degree equations and dynamically separated subtour-elimination constraints (SECs) define each learned transition. At each refinement, the degree equations and selected, strictly satisfied SEC equations, with positive slacks, define an iterate-dependent primal-slack Karush--Kuhn--Tucker (KKT) manifold. Repaired dual variables and violated SEC rows define a local cost field. An exact constrained mirror-descent step maps each finite state to a positive state on the same manifold. Where selected rows and deterministic ties remain fixed, implicit differentiation maps parameter perturbations into the manifold tangent space and reuses the forward constraint operator for the local-cost-field derivative. The terminal edge state allocates computation across Held--Karp ascent, candidate-graph edge tests, and tour construction under a fixed budget. Deterministic verification recomputes original costs and accepts only verified candidate-graph lower bounds and edge decisions. On 1,000 held-out TSP1000 instances, DualCert attains a mean tour-cost gap of \(0.0573\%\) from Lin--Kernighan--Helsgaun version 3 (LKH-3) reference tours in \(9.55\) batch-amortized seconds per instance. It returns a verified candidate-graph lower bound for every instance and achieves \(81.46\%\) edge-decision coverage. The mean gap is \(67.1\%\) smaller than the reported NeuroLKH mean gap. Thus, optimization constraints govern learning, while deterministic verification preserves output validity.

Problem

• Traveling Salesman Problem
• constraint satisfaction
• computation allocation
• output validity
• large-scale optimization

Innovation

• constraint-coupled learning
• KKT manifold
• implicit differentiation
• subtour-elimination constraints
• deterministic verification