The Approximability of the p-hub Center Problem with Parameterized Triangle Inequality

Abstract

A complete weighted graph $G= (V, E, w)$ is called $\varDelta _{\beta }$-metric, for some $\beta \ge 1/2$, if G satisfies the $\beta$-triangle inequality, i.e., $w(u,v) \le \beta \cdot (w(u,x) + w(x,v))$ for all vertices $u,v,x\in V$. Given a $\varDelta _{\beta }$-metric graph $G=(V, E, w)$ and an integer p, the $\varDelta _{\beta }$-p Hub Center Problem ($\varDelta _{\beta }$-pHCP) is to find a spanning subgraph $H^{*}$ of G such that (i) vertices (hubs) in $C^{*}{\subset }V$ form a clique of size p in $H^{*}$; (ii) vertices (non-hubs) in $V{\setminus }C^{*}$ form an independent set in $H^{*}$; (iii) each non-hub $v\in V{\setminus }C^{*}$ is adjacent to exactly one hub in $C^{*}$; and (iv) the diameter $D(H^{*})$ is minimized. For $\beta = 1$, $\varDelta _{\beta }$-pHCP is NP-hard. (Chen et al., CMCT 2016) proved that for any $\varepsilon >0$, it is NP-hard to approximate the $\varDelta _{\beta }$-pHCP to within a ratio $\frac{4}{3}-\varepsilon$ for $\beta = 1$. In the same paper, a $\frac{5}{3}$-approximation algorithm was given for $\varDelta _{\beta }$-pHCP for $\beta = 1$. In this paper, we study $\varDelta _{\beta }$-pHCP for all $\beta \ge \frac{1}{2}$. We show that for any $\varepsilon > 0$, to approximate the $\varDelta _{\beta }$-pHCP to a ratio $g(\beta ) - \varepsilon$ is NP-hard and we give $r(\beta )$-approximation algorithms for the same problem where $g(\beta )$ and $r(\beta )$ are functions of $\beta$. If $\beta \le \frac{3 - \sqrt{3}}{2}$, we have $r(\beta ) = g(\beta ) = 1$, i.e., $\varDelta _{\beta }$-pHCP is polynomial time solvable. If $\frac{3-\sqrt{3}}{2} < \beta \le \frac{2}{3}$, we have $r(\beta ) = g(\beta ) = \frac{3\beta - 2\beta ^2}{3(1-\beta )}$. For $\frac{2}{3} \le \beta \le \frac{5+\sqrt{5}}{10}$, $r(\beta )=g(\beta ) = \beta +\beta ^2$. Moreover, for $\beta \ge 1$, we have $g(\beta ) = \beta \cdot \frac{4\beta - 1}{3\beta -1}$ and $r(\beta ) = 2\beta$, the approximability of the problem (i.e., upper and lower bound) is linear in $\beta$.

Parts of this research were supported by the Ministry of Science and Technology of Taiwan under grants MOST 105–2221–E–006–164–MY3, and MOST 103–2221–E–006–135–MY3.

Li-Hsuan Chen is supported by the Ministry of Science and Technology of Taiwan under grant MOST 106–2811–E–006–008.

Ling-Ju Hung is supported by the Ministry of Science and Technology of Taiwan under grants MOST 105–2811–E–006–046.

Part of this work was done while Ralf Klasing was visiting the Department of Computer Science and Information Engineering at National Cheng Kung University. This study has been carried out in the frame of the “Investments for the future” Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02). Research supported by the LaBRI under the “Projets émergents” program.