In this paper we study the hardness of the k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Center problem on inputs that model transportation networks. For the problem, a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document} with edge lengths and an integer k are given and a center set C⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subseteq V$$\end{document} needs to be chosen such that |C|≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|C|\le k$$\end{document}. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and the pathwidth p. Moreover, under the exponential time hypothesis there is no f(k,p,h)·no(p+k+h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(k,p,h)\cdot n^{o(p+\sqrt{k+h})}$$\end{document} time algorithm for any computable function f. Thus it is unlikely that the optimum solution to k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+{\varepsilon })$$\end{document}-approximation algorithm for inputs of doubling dimension d with runtime (kk/εO(kd))·nO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k^k/{\varepsilon }^{O(kd)})\cdot n^{O(1)}$$\end{document}. This generalizes a previous result, which considered inputs in D-dimensional Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q$$\end{document} metrics.

中文翻译：

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交通网络中 k 中心问题的参数化硬度

在本文中，我们研究了 k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage 的硬度{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} - 交通网络模型输入的中心问题。对于这个问题，图 G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs } \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document} 给出边长和整数 k 和一个中心集 C⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage {amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subseteq V$$\end{document} 需要选择使得 |C|≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|C|\le k$$\end{document}。目的是最小化图中任何顶点到最近中心的最大距离。这个问题出现在物流的许多应用中，因此很自然地考虑对运输网络建模的输入。此类输入通常被假定为平面图、低倍频度量或有界公路维度图。对于这些模型中的每一个，已经证明存在参数化近似算法。我们通过证明 k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{ 来补充这些结果upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-中心问题是 W[1]-在恒定加倍维度的平面图上很难，其中参数是中心数 k、公路尺寸 h 和路径宽度 p 的组合。此外，在指数时间假设下，没有 f(k,p, h)·no(p+k+h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \ usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(k,p,h)\cdot n^{o(p+\sqrt{k+h})}$$\end {document} 任何可计算函数的时间算法 f。因此， k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage 的最佳解决方案不太可能{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k$$\end{document}-Center 可以有效地找到，即使假设输入图符合上述所有模型交通网络一下子！此外，我们给出了一个简单的参数化 (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \ usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+{\varepsilon })$$\end{document} - 输入加倍维度 d 的近似算法和运行时 (kk/ εO(kd))·nO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage {upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k^k/{\varepsilon }^{O(kd)})\cdot n^{O(1)}$$\结束{文档}。这概括了先前的结果，