eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-09-01
30:1
30:14
10.4230/LIPIcs.ESA.2017.30
article
Improving TSP Tours Using Dynamic Programming over Tree Decompositions
Cygan, Marek
Kowalik, Lukasz
Socala, Arkadiusz
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^k). At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2/3k)+1})-time algorithm.
We show an algorithm which runs in O(n^{(1/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 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^{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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol087-esa2017/LIPIcs.ESA.2017.30/LIPIcs.ESA.2017.30.pdf
TSP
treewidth
local search
XP algorithm
hardness in P