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- W4295373208 abstract "We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017]. We show that the LP solution can be sparsified via cut-sparsification techniques such as those of Benczur and Karger [2015]. Given a weighted graph $G$ with $m$ edges and $n$ vertices, and $epsilon > 0$, our randomized algorithm outputs with high probability a $(1+epsilon)$-approximate solution to the LP relaxation whose support has $operatorname{O}(n log n /epsilon^2)$ edges. The running time of the algorithm is $operatorname{~O}(m/epsilon^2)$. This can be generically used to speed up algorithms that rely on the LP. For Metric-TSP, we obtain the following concrete result. For a weighted graph $G$ with $m$ edges and $n$ vertices, and $epsilon > 0$, we describe an algorithm that outputs with high probability a tour of $G$ with cost at most $(1 + epsilon) frac{3}{2}$ times the minimum cost tour of $G$ in time $operatorname{~O}(m/epsilon^2 + n^{1.5}/epsilon^3)$. Previous implementations of Christofides' algorithm [Christofides, 1976] require, for a $frac{3}{2}$-optimal tour, $operatorname{~O}(n^{2.5})$ time when the metric is explicitly given, or $operatorname{~O}(min{m^{1.5}, mn+n^{2.5}})$ time when the metric is given implicitly as the shortest path metric of a weighted graph." @default.
- W4295373208 created "2022-09-13" @default.
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- W4295373208 date "2018-02-04" @default.
- W4295373208 modified "2023-10-18" @default.
- W4295373208 title "Fast Approximations for Metric-TSP via Linear Programming" @default.
- W4295373208 doi "https://doi.org/10.48550/arxiv.1802.01242" @default.
- W4295373208 hasPublicationYear "2018" @default.
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