稀疏有向图中的非对称旅行商问题
The Asymmetric Travelling Salesman Problem in Sparse Digraphs
摘要：
非对称旅行商问题（ATSP）及其特例有向哈密顿性是计算机科学中最基本的问题之一。在时间上运行的动态规划算法$O^*(2^n)$由Bellman、Held和Karp在近60年前开发，至今仍是解决这两个问题的最先进的方法。在这项工作中，我们专注于稀疏有向图。首先，我们回顾已知的方法，在稀疏图中的无向哈密尔顿性和TSP，我们分析其后果的有向哈密尔顿性和ATSP在稀疏有向图，无论是通过调整算法，或通过使用减少。通过这种方法，我们得到了几类稀疏有向图的运行时间上界，其中包括出度和入度均以2为界的有向图的$O^*(2^{n/3})$和出度以3为界的有向图的$O^*(3^{n/2})$。我们的主要结果集中在有界平均出度$d$的有向图上。这里ATSP的基线是一个简单的循环覆盖枚举，可以在函数$μ(d)\le(\lceil{d}\rceil!)^{1/{\lceil{d}\rceil}}$的$O^*(μ(d)^n)$的时间范围内完成。人们还可以观察到，定向哈密顿性可以在随机时间$O^*((2-2^{-d})^n)$和多项式空间中求解，通过调整Björklund [ISAAC 2018]的最新结果，最初用于稀疏二分图中的无向哈密顿性。我们提出了两个新的确定性算法的ATSP：第一个运行在时间$O(2^{0.441(d-1)n})$和多项式空间，第二个在指数空间与运行时间的$O^*(τ(d)^{n/2})$的功能$τ(d)\le d$。
英文摘要：
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 $O^*(μ(d)^n)$ for a function $μ(d)\le(\lceil{d}\rceil!)^{1/{\lceil{d}\rceil}}$. 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 Björklund [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 $O^*(τ(d)^{n/2})$ for a function $τ(d)\le d$.
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