2,203 research outputs found

    The Salesman's Improved Tours for Fundamental Classes

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    Finding the exact integrality gap α\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 4/3α3/24/3 \leq \alpha \leq 3/2, and a famous conjecture states α=4/3\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 ρ\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 1/21/2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3/23/2 to 10/710/7 for a superclass of these points, as well as prove a lower bound of 4/34/3 for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

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

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    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)G=(V,E) has an α\alpha-uniform cover for TSP (2EC, respectively) if the everywhere α\alpha vector (i.e. {α}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 (1ϵ)(1-\epsilon)-uniform covers for TSP for some ϵ>0\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

    A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case

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    Given a connected undirected graph Gˉ\bar{G} on nn vertices, and non-negative edge costs cc, the 2ECM problem is that of finding a 22-edge~connected spanning multisubgraph of Gˉ\bar{G} of minimum cost. The natural linear program (LP) for 2ECM, which coincides with the subtour LP for the Traveling Salesman Problem on the metric closure of Gˉ\bar{G}, gives a lower bound on the optimal cost. For instances where this LP is optimized by a half-integral solution xx, Carr and Ravi (1998) showed that the integrality gap is at most 43\frac43: they show that the vector 43x\frac43 x dominates a convex combination of incidence vectors of 22-edge connected spanning multisubgraphs of Gˉ\bar{G}. We present a simpler proof of the result due to Carr and Ravi by applying an extension of Lov\'{a}sz's splitting-off theorem. Our proof naturally leads to a 43\frac43-approximation algorithm for half-integral instances. Given a half-integral solution xx to the LP for 2ECM, we give an O(n2)O(n^2)-time algorithm to obtain a 22-edge connected spanning multisubgraph of Gˉ\bar{G} whose cost is at most 43cTx\frac43 c^T x

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

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    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 θ\theta-cyclic point: xe{0,θ,1θ,1}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 xx-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 xx-value 1 from the 32\frac{3}{2} of Christofides algorithm to 32θ10\frac{3}{2}-\frac{\theta}{10} while keeping the usage of edges with fractional xx-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 23\frac{2}{3}-uniform point: xe{0,23}x_e \in \{0, \frac{2}{3}\}, we give a 1712\frac{17}{12}-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios of 32\frac{3}{2} of Christofides algorithm and 43\frac{4}{3} implied by the famous "four-thirds conjecture"
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