133 research outputs found

    Approximating the Regular Graphic TSP in near linear time

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    We present a randomized approximation algorithm for computing traveling salesperson tours in undirected regular graphs. Given an nn-vertex, kk-regular graph, the algorithm computes a tour of length at most (1+7lnkO(1))n\left(1+\frac{7}{\ln k-O(1)}\right)n, with high probability, in O(nklogk)O(nk \log k) time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS 2012) for the same problem, in terms of both approximation factor, and running time. The key ingredient of our algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with O(n/logk)O(n/\log k) cycles with high probability, in near linear time. Additionally, we also give a deterministic 32+O(1k)\frac{3}{2}+O\left(\frac{1}{\sqrt{k}}\right) factor approximation algorithm running in time O(nk)O(nk).Comment: 12 page

    A 9/7 -Approximation Algorithm for Graphic TSP in Cubic Bipartite Graphs

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    We prove new results for approximating Graphic TSP. Specifically, we provide a polynomial-time 9/7-approximation algorithm for cubic bipartite graphs and a (9/7+1/(21(k-2)))-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with our knowledge of the structure of cubic graphs

    Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation

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    We consider the problem of designing sublinear time algorithms for estimating the cost of a minimum metric traveling salesman (TSP) tour. Specifically, given access to a n×nn \times n distance matrix DD that specifies pairwise distances between nn points, the goal is to estimate the TSP cost by performing only sublinear (in the size of DD) queries. For the closely related problem of estimating the weight of a metric minimum spanning tree (MST), it is known that for any ε>0\varepsilon > 0, there exists an O~(n/εO(1))\tilde{O}(n/\varepsilon^{O(1)}) time algorithm that returns a (1+ε)(1 + \varepsilon)-approximate estimate of the MST cost. This result immediately implies an O~(n/εO(1))\tilde{O}(n/\varepsilon^{O(1)}) time algorithm to estimate the TSP cost to within a (2+ε)(2 + \varepsilon) factor for any ε>0\varepsilon > 0. However, no o(n2)o(n^2) time algorithms are known to approximate metric TSP to a factor that is strictly better than 22. On the other hand, there were also no known barriers that rule out the existence of (1+ε)(1 + \varepsilon)-approximate estimation algorithms for metric TSP with O~(n)\tilde{O}(n) time for any fixed ε>0\varepsilon > 0. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost. We also show that the problem of estimating metric TSP cost is closely connected to the problem of estimating the size of a maximum matching in a graph.Comment: ICALP 202

    The traveling salesman problem on cubic and subcubic graphs

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    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
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