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On the Complexity of the Star p-hub Center Problem with Parameterized Triangle Inequality

  • Li-Hsuan Chen
  • Sun-Yuan Hsieh
  • Ling-Ju HungEmail author
  • Ralf Klasing
  • Chia-Wei Lee
  • Bang Ye Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

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 a center \(c\in V\), and an integer p, the \(\varDelta _{\beta }\)-Star p-Hub Center Problem (\(\varDelta _{\beta }\)-SpHCP) is to find a depth-2 spanning tree T of G rooted at c such that c has exactly p children and the diameter of T is minimized. The children of c in T are called hubs. For \(\beta = 1\), \(\varDelta _{\beta }\)-SpHCP is NP-hard. (Chen et al., COCOON 2016) proved that for any \(\varepsilon >0\), it is NP-hard to approximate the \(\varDelta _{\beta }\)-SpHCP to within a ratio \(1.5-\varepsilon \) for \(\beta = 1\). In the same paper, a \(\frac{5}{3}\)-approximation algorithm was given for \(\varDelta _{\beta }\)-SpHCP for \(\beta = 1\). In this paper, we study \(\varDelta _{\beta }\)-SpHCP for all \(\beta \ge \frac{1}{2}\). We show that for any \(\varepsilon > 0\), to approximate the \(\varDelta _{\beta }\)-SpHCP 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., \(\varDelta _{\beta }\)-SpHCP is polynomial time solvable. If \(\frac{3-\sqrt{3}}{2} < \beta \le \frac{2}{3}\), we have \(r(\beta ) = g(\beta ) = \frac{1 + 2\beta - 2\beta ^2}{4(1-\beta )}\). For \(\frac{2}{3} \le \beta \le 1\), \(r(\beta ) = \min \{\frac{1 + 2\beta - 2\beta ^2}{4(1-\beta )}, 1 + \frac{4\beta ^2}{5\beta +1}\}\). Moreover, for \(\beta \ge 1\), we have \(r(\beta ) = \min \{\beta + \frac{4\beta ^2- 2\beta }{2 + \beta }, 2\beta + 1\}\). For \(\beta \ge 2\), the approximability of the problem (i.e., upper and lower bound) is linear in \(\beta \).

Keywords

Approximation Algorithm Polynomial Time Triangle Inequality Approximation Ratio Travel Salesman Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Li-Hsuan Chen
    • 1
  • Sun-Yuan Hsieh
    • 1
  • Ling-Ju Hung
    • 1
    Email author
  • Ralf Klasing
    • 2
  • Chia-Wei Lee
    • 1
  • Bang Ye Wu
    • 3
  1. 1.Department of Computer Science and Information EngineeringNational Cheng Kung UniversityTainanTaiwan
  2. 2.CNRS, LaBRIUniversité de BordeauxTalence CedexFrance
  3. 3.Department of Computer Science and Information EngineeringNational Chung Cheng UniversityChiayiTaiwan

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