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Approximation and Parameterized Algorithms for Covering with Disks of Two Types of Radii

Authors Sayan Bandyapadhyay , Eli Mitchell

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Approximation algorithms
Keywords
  • Covering
  • Disks
  • Approximation
  • FPT

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Abstract

We study the Discrete Covering with Two Types of Radii problem motivated by its application in wireless networks. In this problem, the goal is to assign either small-range high frequency or large-range low frequency to each access point, maximizing the number of users in high-frequency regions while ensuring that each user is in the range of an access point. Unlike other weighted covering problems, our problem requires satisfying two simultaneous objectives, which calls for novel approaches that leverage the underlying geometry of the problem. In our work, we present two new algorithms: the first is a polynomial-time (2.5 + ε)-approximation, and the second is an exact algorithm for sparse instances, which is fixed-parameter tractable (FPT) in the number of large-radius disks. We also prove that such an FPT algorithm is impossible for general instances lacking sparsity, assuming the Exponential Time Hypothesis. Before our work, the best-known polynomial-time approximation factor was 4 for the problem. 
Our approximation algorithm results from a fine-grained classification of points that can contribute to the gain of a solution. Based on this classification, we design two sub-algorithms with interdependent guarantees to recover the respective class of points as gain. Our algorithm exploits further properties of Delaunay triangulations to achieve the improved bound. The FPT algorithm is based on branching that utilizes the sparsity of the instances to limit the overall search space.

Cite As Get BibTex

Sayan Bandyapadhyay and Eli Mitchell. Approximation and Parameterized Algorithms for Covering with Disks of Two Types of Radii. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.7

Author Details

Sayan Bandyapadhyay
  • Portland State University, OR, USA
Eli Mitchell
  • Department of Computer Science, Portland State University, OR, USA

Funding

  • Bandyapadhyay, Sayan: The work has been supported by the NSF grant 2311397.

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