The Asymmetric Travelling Salesman Problem in Sparse Digraphs

Lukasz Kowalik, Konrad Majewski
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引用次数: 1

Abstract

Asymmetric Travelling Salesman Problem (ATSP) and its special case Directed Hamiltonicity are among the most fundamental problems in computer science. The dynamic programming algorithm running in time $O^*(2^n)$ developed almost 60 years ago by Bellman, Held and Karp, is still the state of the art for both of these problems. In this work we focus on sparse digraphs. First, we recall known approaches for Undirected Hamiltonicity and TSP in sparse graphs and we analyse their consequences for Directed Hamiltonicity and ATSP in sparse digraphs, either by adapting the algorithm, or by using reductions. In this way, we get a number of running time upper bounds for a few classes of sparse digraphs, including $O^*(2^{n/3})$ for digraphs with both out- and indegree bounded by 2, and $O^*(3^{n/2})$ for digraphs with outdegree bounded by 3. Our main results are focused on digraphs of bounded average outdegree $d$. The baseline for ATSP here is a simple enumeration of cycle covers which can be done in time bounded by $O^*(\mu(d)^n)$ for a function $\mu(d)\le(\lceil{d}\rceil!)^{1/{\lceil{d}\rceil}}$. One can also observe that Directed Hamiltonicity can be solved in randomized time $O^*((2-2^{-d})^n)$ and polynomial space, by adapting a recent result of Bjorklund [ISAAC 2018] stated originally for Undirected Hamiltonicity in sparse bipartite graphs. We present two new deterministic algorithms for ATSP: the first running in time $O(2^{0.441(d-1)n})$ and polynomial space, and the second in exponential space with running time of $O^*(\tau(d)^{n/2})$ for a function $\tau(d)\le d$.
稀疏有向图中的不对称旅行商问题
不对称旅行商问题(ATSP)及其特例有向哈密性是计算机科学中最基本的问题之一。由Bellman, Held和Karp在近60年前开发的动态规划算法$O^*(2^n)$在这两个问题上仍然是最先进的。在这项工作中,我们关注稀疏有向图。首先,我们回顾了稀疏图中的无向哈密性和TSP的已知方法,并分析了它们对稀疏有向图中的有向哈密性和ATSP的影响,无论是通过调整算法,还是通过使用约简。通过这种方式,我们得到了一些稀疏有向图类的运行时间上限,包括出度和度均为2的有向图的$O^*(2^{n/3})$,出度均为3的有向图的$O^*(3^{n/2})$。我们的主要结果集中在有界平均度的有向图$d$上。这里,ATSP的基线是一个简单的循环覆盖枚举,对于一个函数$\mu(d)\le(\lceil{d}\rceil!)^{1/{\lceil{d}\rceil}}$,它可以在以$O^*(\mu(d)^n)$为界的时间内完成。我们还可以观察到,有向哈密性可以在随机时间$O^*((2-2^{-d})^n)$和多项式空间中求解,通过采用Bjorklund [ISAAC 2018]最初对稀疏二部图中的无向哈密性提出的最新结果。我们提出了两个新的ATSP确定性算法:第一个是在时间$O(2^{0.441(d-1)n})$和多项式空间中运行,第二个是在函数$\tau(d)\le d$的指数空间中运行,运行时间$O^*(\tau(d)^{n/2})$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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