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 Salesman's Improved Tours for Fundamental Classes

Finding the exact integrality gap $\alpha$ for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that $4/3 \leq \alpha \leq 3/2$, and a famous conjecture states $\alpha = 4/3$. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most $\rho$ for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the $1/2$-integer points of one such class. We successfully improve the upper bound for the integrality gap from $3/2$ to $10/7$ for a superclass of these points, as well as prove a lower bound of $4/3$ for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

Finding a Maximum 2-Matching Excluding Prescribed Cycles in Bipartite Graphs

We introduce a new framework of restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if, for each setU in U, F contains at most |setU|-1 edges in the subgraph induced by U. Our framework includes C_{= 5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C_{<=6}-free 2-matching problem in O(n^2 m) time, where n and m are the numbers of vertices and edges, respectively

Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a $\theta$-cyclic point: $x_e \in \{0,\theta, 1-\theta, 1\}$, where the support graph is subcubic and each vertex is incident to at least one edge with $x$-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with $x$-value 1 from the $\frac{3}{2}$ of Christofides algorithm to $\frac{3}{2}-\frac{\theta}{10}$ while keeping the usage of edges with fractional $x$-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a $\frac{2}{3}$-uniform point: $x_e \in \{0, \frac{2}{3}\}$, we give a $\frac{17}{12}$-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios of $\frac{3}{2}$ of Christofides algorithm and $\frac{4}{3}$ implied by the famous "four-thirds conjecture"

A hierarchical approach to improve the ant colony optimization algorith

The ant colony optimization algorithm (ACO) is a fast heuristic-based method for finding favorable solutions to the traveling salesman problem (TSP). When the data set reaches larger values however, the ACO runtime increases dramatically. As a result, clustering nodes into groups is an effective way to reduce the size of the problem while leveraging the advantages of the ACO algorithm. The method for recombining groups of nodes is explored by treating the graph as a hierarchy of clusters, and modifying the original ACO heuristic to operate on a hypergraph. This method of using hierarchical clustering is significantly faster than the original ACO algorithm, even when normal clustering techniques are applied, while producing improved tour lengths

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