The asymmetric p-center problem (ApCP) was proved by Chuzhoy et al. (STOC’04) to be NP-hard to approximate within a factor of \(\log ^*n - \Theta (1)\) unless \(\mathrm {NP} \subseteq \mathrm {DTIME}(n^{\log \log n})\). This paper studies ApCP and the vertex-weighted asymmetric p-center problem (WApCP). First, we propose four classes of parameterized complete digraphs, \(\alpha \)-CD, \((\alpha , \beta )\)-CD, \(\langle \alpha , \gamma \rangle \)-CD and \((\alpha , \beta , \gamma )\)-CD, from the angle of the parameterized upper bound on the ratio of two asymmetric edge weights between vertices as well as on the ratio of two vertex weights, and the parameterized triangle inequality, respectively. Using the greedy approach, we achieve a \((1 + \alpha )\)- and \(\beta \cdot (1 + \alpha )\)-approximation algorithm for the ApCP in \(\alpha \)-CD’s and \((\alpha , \beta )\)-CD’s, respectively, as well as a \((1 + \alpha \gamma )\)- and \(\beta \cdot (1 + \alpha \gamma )\)-approximation algorithm for the WApCP in \(\langle \alpha , \gamma \rangle \)-CD’s and \((\alpha , \beta , \gamma )\)-CD’s, respectively.

中文翻译：

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在参数化有向图中逼近非对称p中心问题

Chuzhoy等人证明了非对称p中心问题（A p CP）。（STOC'04）是NP硬的因子内近似\（\登录^ * N - \西塔（1）\）除非\（\ mathrm {NP} \ subseteq \ mathrm {DTIME}（N ^ { \ log \ log n}）\）。本文研究A p CP和顶点加权不对称p中心问题（WA p CP）。首先，我们提出四类参数化的完整有向图，\（\ alpha \）- CD，\（（\ alpha，\ beta）\）- CD，\（\ langle \ alpha，\ gamma \ rangle \）- CD和\（（\ alpha，\ beta，\ gamma）\）-CD，分别从两个顶点之间的两个不对称边权重之比，两个顶点权重之比以及参数化的三角形不等式的参数上限的角度出发。使用贪婪方法，我们实现了\（（1 + \ alpha）\） -和\（\ beta \ cdot（1 + \ alpha）\） - \（\ alpha \）中A p CP的近似算法- CD和\（（\ alpha，\ beta）\）- CD，以及\（（1 + \ alpha \ gamma）\） -和\（\ beta \ cdot（1 + \ alpha \ gamma） \） - （\ langle \ alpha，\ gamma \ rangle \）- CD's中WA p CP的近似算法\（（（\ alpha，\ beta，\ gamma）\）- CD。