On the PLS-Completeness of $k$-Opt Local Search for the Traveling Salesman Problem

2026-03-11

AI Summary

This study investigates the computational complexity of $k$-Opt local search for the Traveling Salesman Problem (TSP), aiming to identify the smallest $k$ for which $k$-Opt is PLS-complete. By constructing a refined graph reduction, the paper establishes the first rigorous proof that $k$-Opt is PLS-complete for both general and metric TSP when $k \geq 15$, dramatically lowering the previously known threshold of $k \gg 1000$. This result not only closes a critical gap in prior proofs but also resolves a long-standing open question in the theoretical analysis of local search algorithms, thereby providing a stronger foundation for understanding the complexity landscape of TSP heuristics.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The $k$-Opt algorithm is a local search algorithm for the traveling salesman problem. Starting with an initial tour, it iteratively replaces at most $k$ edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the traveling salesman problem with the $k$-Opt neighborhood is complete for the class PLS (polynomial time local search). However, his proof requires $k \gg 1000$ and has a substantial gap. We provide the first rigorous proof for the PLS-completeness and at the same time drastically lower the value of $k$ to $k \geq 15$, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). Our result holds for both the general and the metric traveling salesman problem.

Problem

• k-Opt
• Traveling Salesman Problem
• PLS-completeness
• local search
• computational complexity

Innovation

• k-Opt
• PLS-completeness
• Traveling Salesman Problem
• local search
• computational complexity