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Hardness and Approximation for the Star \(\beta \)-Hub Routing Cost Problem in \(\varDelta _\beta \)-Metric Graphs

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Computing and Combinatorics (COCOON 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14422))

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Abstract

Minimizing transportation costs through the design of a hub-and-spoke network is a crucial concern in hub location problems (HLP). Within the realm of HLP, the \(\varDelta _\beta \) -Star p -Hub Routing Cost Problem (\(\varDelta _\beta \)-SpHRP) represents an open problem stemming from the Star p -Hub Routing Cost Problem (SpHRP) discussed in a publication by [Yeh et al., Theoretical Computer Science, 2022]. The \(\varDelta _\beta \)-SpHRP deals with a specific vertex c, a positive integer p, and a \(\varDelta _\beta \)-metric graph denoted as G, which is an undirected, complete, and weighted graph adhering to the \(\beta \)-triangle inequality. The objective is to identify a spanning tree T that satisfies the following conditions: it is rooted at c, contains exactly p hubs adjacent to c, and assigns each remaining vertex to a hub while minimizing the routing cost of T. This paper expands the input instances from metric graphs to \(\varDelta _{\beta }\)-metric graphs. Our research demonstrates that SpHRP is NP-hard for any \(\beta >\frac{1}{2}\), indicating that SpHRP remains NP-hard for various subclasses of metric graphs. For approximation algorithms, we introduce two approaches that improve upon previous results, particularly when \(\beta \) is close to \(\frac{1}{2}\).

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan

    Meng-Shiou Tsai & Sun-Yuan Hsieh

  2. Department of Creative Technologies and Product Design, National Taipei University of Business, Taoyuan City, Taiwan

    Ling-Ju Hung

Authors
  1. Meng-Shiou Tsai
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  2. Sun-Yuan Hsieh
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  3. Ling-Ju Hung
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Correspondence to Ling-Ju Hung .

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Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. University of Delaware, Newark, DE, USA

    Guangmo Tong

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Tsai, MS., Hsieh, SY., Hung, LJ. (2024). Hardness and Approximation for the Star \(\beta \)-Hub Routing Cost Problem in \(\varDelta _\beta \)-Metric Graphs. In: Wu, W., Tong, G. (eds) Computing and Combinatorics. COCOON 2023. Lecture Notes in Computer Science, vol 14422. Springer, Cham. https://doi.org/10.1007/978-3-031-49190-0_7

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  • DOI: https://doi.org/10.1007/978-3-031-49190-0_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49189-4

  • Online ISBN: 978-3-031-49190-0

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