A near-complete resolution of the exponential-time complexity of k-opt for the traveling salesman problem

2025-07-16

AI Summary

This paper investigates the iteration complexity of the *k*-opt algorithm for the Traveling Salesman Problem (TSP) under the best-improvement pivot rule. For the long-standing open cases *k* = 3 and *k* = 4, we construct explicit counterexamples that rigorously establish exponential lower bounds on the number of iterations required—both in general and metric TSP instances—thereby resolving the fundamental question of asymptotic lower bounds for *k*-opt when *k* ≥ 3. We further extend these results to the 2.5-opt algorithm. Methodologically, our approach integrates combinatorial construction, graph-theoretic modeling, and rigorous complexity analysis; carefully engineered instances expose the structural mechanism by which local search can be trapped in exponentially long improvement sequences. This work delivers the first exponential lower bounds for *k*-opt with *k* = 3, *k* = 4, and 2.5-opt under the best-improvement rule, filling a critical theoretical gap and providing foundational insights into the worst-case behavior of local search algorithms for TSP.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The $k$-opt algorithm is one of the simplest and most widely used heuristics for solving the traveling salesman problem. Starting from an arbitrary tour, the $k$-opt algorithm improves the current tour in each iteration by exchanging up to $k$ edges. The algorithm continues until no further improvement of this kind is possible. For a long time, it remained an open question how many iterations the $k$-opt algorithm might require for small values of $k$, assuming the use of an optimal pivot rule. In this paper, we resolve this question for the cases $k = 3$ and $k = 4$ by proving that in both these cases an exponential number of iterations may be needed even if an optimal pivot rule is used. Combined with a recent result from Heimann, Hoang, and Hougardy (ICALP 2024), this provides a complete answer for all $k geq 3$ regarding the number of iterations the $k$-opt algorithm may require under an optimal pivot rule. In addition we establish an analogous exponential lower bound for the 2.5-opt algorithm, a variant that generalizes 2-opt and is a restricted version of 3-opt. All our results hold for both the general and the metric traveling salesman problem.

Problem

• Resolves exponential-time complexity of k-opt for TSP
• Determines iteration limits for k=3 and k=4 cases
• Extends results to 2.5-opt algorithm variants

Innovation

• Resolves exponential-time complexity for k-opt
• Proves exponential iterations for k=3 and k=4
• Establishes lower bound for 2.5-opt algorithm