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

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.