Approximating the asymmetric p-center problem in parameterized complete digraphs

Abstract

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.