从中国邮递员到推销员及其超越II：不可近似性与参数化复杂性
From Chinese Postman to Salesman and Beyond II: Inapproximability and Parameterized Complexity
AI总结：
本文研究连续图模型中的$δ$-Tour问题，证明了对于$δ<3/2$的APX难性和FPT性，对于$δ\geq3/2$的强不可近似性和W[2]难性，并给出了$δ$作为输入时的精确算法及其基于ETH的下界。
摘要：
一个被广泛研究的图的连续模型将每条边视为一个连续的单位长度区间点集。在该模型中定义的问题$δ$-Tour，目标是找到一条最短的游览路线，使其与每条边上的每个点的距离不超过$δ$。这个参数化问题在本文的前身中被引入，并证明对于$δ= 0$，它本质上等价于中国邮递员问题；对于$δ= 1/2$，等价于图形旅行推销员问题（TSP）；对于更大的$δ$，则接近先是最小顶点覆盖问题，然后是支配集问题。此外，之前还提供了多个参数范围的近似算法。在本文中，我们补充了不可近似性下界，并研究了该问题的固定参数可解性。一方面，我们证明以下结果：（1）对于每个固定的$0 < δ< 3/2$，问题$δ$-Tour是APX难的，而对于每个固定的$δ\geq 3/2$，除非P = NP，否则该问题没有多项式时间的$o(\log{n})$近似算法。我们的技术还产生了新结果：TSP在三次图（甚至三次二分图）上仍然是APX难的。（2）对于每个固定的$0 < δ< 3/2$，当以最短游览长度为参数时，问题$δ$-Tour是固定参数可解的（FPT），而对于每个固定的$δ\geq 3/2$，它是W[2]难的，并且当$δ$作为输入的一部分时是para-NP难的。另一方面，如果$δ$被视为输入的一部分，那么当$δ$是顶点数的常数比例时，会出现一个有趣的非平凡现象：（3）如果$δ$是输入的一部分，那么该问题可以在时间$f(k)n^{O(k)}$内解决，其中$k = \lceil n/δ\rceil$；然而，假设指数时间假说（ETH），不存在在时间$f(k)n^{o(k/\log k)}$内解决该问题的算法。
英文摘要：
A well-studied continuous model of graphs considers each edge as a continuous unit-length interval of points. In the problem $δ$-Tour defined within this model, the objective to find a shortest tour that comes within a distance of $δ$ of every point on every edge. This parameterized problem was introduced in the predecessor to this article and shown to be essentially equivalent to the Chinese Postman problem for $δ= 0$, to the graphic Travel Salesman Problem (TSP) for $δ= 1/2$, and close to first Vertex Cover and then Dominating Set for even larger $δ$. Moreover, approximation algorithms for multiple parameter ranges were provided. In this article, we provide complementing inapproximability bounds and examine the fixed-parameter tractability of the problem. On the one hand, we show the following: (1) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour is APX-hard, while for every fixed $δ\geq 3/2$, the problem has no polynomial-time $o(\log{n})$-approximation unless P = NP. Our techniques also yield the new result that TSP remains APX-hard on cubic (and even cubic bipartite) graphs. (2) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour is fixed-parameter tractable (FPT) when parameterized by the length of a shortest tour, while it is W[2]-hard for every fixed $δ\geq 3/2$ and para-NP-hard for $δ$ being part of the input. On the other hand, if $δ$ is considered to be part of the input, then an interesting nontrivial phenomenon occurs when $δ$ is a constant fraction of the number of vertices: (3) If $δ$ is part of the input, then the problem can be solved in time $f(k)n^{O(k)}$, where $k = \lceil n/δ\rceil$; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time $f(k)n^{o(k/\log k)}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
