Finding the Best 3-{OPT} Move in Subcubic Time

Given a Traveling Salesman Problem solution, the best 3-OPT move requires us to remove three edges and replace them with three new ones so as to shorten the tour as much as possible. No worst-case algorithm better than the \u398(n3 ) enumeration of all triples is likely to exist for this problem, but algorithms with average case O(n3 12\u25b ) are not ruled out. In this paper we describe a strategy for 3-OPT optimization which can find the best move by looking only at a fraction of all possible moves. We extend our approach also to some other types of cubic moves, such as some special 6-OPT and 5-OPT moves. Empirical evidence shows that our algorithm runs in average subcubic time (upper bounded by O(n2.5 )) on a wide class of random graphs as well as Traveling Salesman Problem Library (TSPLIB) instances

Shorter tours and longer detours: Uniform covers and a bit beyond

Motivated by the well known four-thirds conjecture for the traveling salesman problem (TSP), we study the problem of {\em uniform covers}. A graph $G=(V,E)$ has an $\alpha$-uniform cover for TSP (2EC, respectively) if the everywhere $\alpha$ vector (i.e. $\{\alpha\}^{E}$) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides' algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Seb\H{o} asked if such graphs have $(1-\epsilon)$-uniform covers for TSP for some $\epsilon > 0$. Indeed, the four-thirds conjecture implies that such graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere 15/17 vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere 2/3 vector is an optimal solution for the subtour linear programming relaxation, then a tour with weight at most 27/19 times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose an optimal solution for the subtour relaxation for TSP into spanning, connected multigraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a 17/12-approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs

The traveling salesman problem on cubic and subcubic graphs

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on TeX vertices a tour of length TeX exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Mömke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs