4,468 research outputs found

    Approximating ATSP by Relaxing Connectivity

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    The standard LP relaxation of the asymmetric traveling salesman problem has been conjectured to have a constant integrality gap in the metric case. We prove this conjecture when restricted to shortest path metrics of node-weighted digraphs. Our arguments are constructive and give a constant factor approximation algorithm for these metrics. We remark that the considered case is more general than the directed analog of the special case of the symmetric traveling salesman problem for which there were recent improvements on Christofides' algorithm. The main idea of our approach is to first consider an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. For this relaxed problem, it is quite easy to give an algorithm with a guarantee of 3 on node-weighted shortest path metrics. More surprisingly, we then show that any algorithm (irrespective of the metric) for the relaxed problem can be turned into an algorithm for the asymmetric traveling salesman problem by only losing a small constant factor in the performance guarantee. This leaves open the intriguing task of designing a "good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio

    Subgraphs in preferential attachment models

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    We consider subgraph counts in general preferential attachment models with power-law degree exponent Ï„>2\tau>2. For all subgraphs HH, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a P\'olya urn model
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