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arXiv 1702.04307cs.DS
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度量TSP的Held-Karp界在近似线性时间上的逼近

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

Chandra Chekuri, Kent Quanrud

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摘要：

我们给出了度量TSP的Held-Karp界[Held和Karp，1970]的一个近似线性时间随机逼近方案。形式上，给定$m$边和$ε> 0$上的无向边加权图$G$，该算法在$O(m \log^4n /ε^2)$时间内以高概率输出$(1+ε)$ -近似，该近似是由$G$上的最短路径度量引起的度量TSP实例上的Held-Karp界。该算法还可以用于输出对应的解到Subtour Elimination LP。我们大大提高了$O(m^2 \log^2(m)/ε^2)$的运行时间之前实现的Garg和Khandekar。LP解可用于获得度量TSP的快速随机化$\big(\frac{3}{2} + ε\big)$ -近似，其改进了Christofides算法的先前实现的运行时间。

英文摘要：

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $ε> 0$, the algorithm outputs in $O(m \log^4n /ε^2)$ time, with high probability, a $(1+ε)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/ε^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + ε\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm.

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