Improving TSP tours using dynamic programming over tree decomposition

TL;DR

This paper introduces a new dynamic programming algorithm over tree decompositions that significantly improves the efficiency of finding improving k-moves in TSP heuristics for k=5 to 10, outperforming previous methods.

Contribution

The authors present a novel algorithm with a runtime of O(n^{(1/4+ε_k)k}) for finding improving k-moves in TSP, surpassing prior algorithms for k=5 to 10, and provide a refined approach for k=5.

Findings

New algorithm runs in O(n^{(1/4+ε_k)k}) time, improving over previous methods for k=5 to 10.

For k=5, the algorithm achieves a runtime of O(n^{3.4}).

Improving the k=4 case would imply breakthroughs in shortest path algorithms.

Abstract

Given a traveling salesman problem (TSP) tour $H$ in graph $G$ a $k$-move is an operation which removes $k$ edges from $H$, and adds $k$ edges of $G$ so that a new tour $H'$ is formed. The popular $k$-OPT heuristics for TSP finds a local optimum by starting from an arbitrary tour $H$ and then improving it by a sequence of $k$-moves. Until 2016, the only known algorithm to find an improving $k$-move for a given tour was the naive solution in time $O(n^k)$. At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an $O(n^{\lfloor 2/3k \rfloor+1})$-time algorithm. We show an algorithm which runs in $O(n^{(1/4+\epsilon_k)k})$ time, where $\lim \epsilon_k = 0$. We are able to show that it improves over the state of the art for every $k=5,\ldots,10$. For the most practically relevant case $k=5$ we provide a slightly refined algorithm running in $O(n^{3.4})$ time. We also show that for the $k=4$ case, improving over the $O(n^3)$-time algorithm of de Berg et al. would be a major breakthrough: an $O(n^{3-\epsilon})$-time algorithm for any $\epsilon>0$ would imply an $O(n^{3-\delta})$-time algorithm for the ALL PAIRS SHORTEST PATHS problem, for some $\delta>0$.