Improving TSP Tours Using Dynamic Programming over Tree Decompositions

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 heuristic 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). At ICALP’16 de Berg, Buchin, Jansen and Woeginger showed an O(n)-time algorithm. We show an algorithm which runs in O(nk) time, where limk→∞ ǫk = 0. It improves over the state of the art for every k ≥ 5. For the most practically relevant case k = 5 we provide a slightly refined algorithm running in O(n) time. We also show that for the k = 4 case, improving over the O(n)-time algorithm of de Berg et al. would be a major breakthrough: an O(n)-time algorithm for any ǫ > 0 would imply an O(n)-time algorithm for the All Pairs Shortest Paths problem, for some δ > 0.