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A QPTAS for Facility Location on Unit Disk Graphs

Authors Zachary Friggstad , Mohsen Rezapour , Mohammad R. Salavatipour , Hao Sun

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Facility Location
  • Unit Disk Graphs
  • Approximation Algorithms

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Abstract

We study the classic (Uncapacitated) Facility Location problem on Unit Disk Graphs (UDGs). For a given point set P in the plane, the unit disk graph UDG(P) on P has vertex set P and an edge between two distinct points p, q ∈ P if and only if their Euclidean distance |pq| is at most 1. The weight of the edge pq is equal to their distance |pq|. An instance of {Facility Location} on UDG(P) consists of a set C ⊆ P of clients and a set F ⊆ P of facilities, each having an opening cost f_i. The goal is to pick a subset F' ⊆ F to open while minimizing ∑_{i ∈ F'} f_i + ∑_{v ∈ C} d(v,F'), where d(v,F') is the distance of v to nearest facility in F' through UDG(P).
In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.

Cite As Get BibTex

Zachary Friggstad, Mohsen Rezapour, Mohammad R. Salavatipour, and Hao Sun. A QPTAS for Facility Location on Unit Disk Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.27

Author Details

Zachary Friggstad
  • Department of Computing Science, University of Alberta, Edmonton, Canada
Mohsen Rezapour
  • Department of Computing Science, University of Alberta, Edmonton, Canada
Mohammad R. Salavatipour
  • Department of Computing Science, University of Alberta, Edmonton, Canada
Hao Sun
  • Department of Computer Science, University of Houston, TX, USA

Funding

  • Friggstad, Zachary: Supported by NSERC.
  • Salavatipour, Mohammad R.: Supported by NSERC.

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