Improving TSP tours using dynamic programming over tree decomposition

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 $H^{\prime }$ 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 $k=5,\dots ,10$. 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 $\delta >0$.

Accepted to ESA'17