We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a small constant factor in the performance guarantee. Next, we present a reduction from general ATSP instances to structured instances, on which we then solve Subtour Partition Cover, yielding our constant-factor approximation algorithm for ATSP.

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一种非对称旅行商问题的常数因子逼近算法

我们给出了一个非对称旅行商问题（ATSP）的常数因子逼近算法。我们的近似保证是针对标准 LP 松弛进行分析的，因此我们的结果证实了该松弛的猜想恒定完整性间隙。我们方法的主要思想是减少 Subtour Partition Cover，这是一个通过将一般连接要求显着放宽到本地连接条件来获得的更简单的问题。我们首先表明，任何 Subtour Partition Cover 算法都可以转换为 ATSP 算法，同时在性能保证中只损失一个小的常数因子。接下来，我们提出了从一般 ATSP 实例到结构化实例的缩减，然后我们在其上求解 Subtour Partition Cover，产生我们的 ATSP 常数因子逼近算法。