Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

A. Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, Shakhar Smorodinsky

Research output: Contribution to journalArticlepeer-review


We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and a fixed parameter d≥ 1 , in the maximum diameter-bounded subgraph problem (MaxDBS for short) the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For d= 1 , this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor n1-ϵ, for any ϵ> 0. Moreover, it is known that, for any d≥ 2 , it is NP-hard to approximate MaxDBS within a factor n1/2-ϵ, for any ϵ> 0. In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems, and several geometric properties of unit disk graphs.

Original languageEnglish
Pages (from-to)1401-1414
Number of pages14
JournalDiscrete and Computational Geometry
Issue number4
StatePublished - 1 Dec 2021


  • Approximation algorithms
  • Fractional Helly theorem
  • Maximum diameter-bounded subgraph
  • Unit disk graphs
  • VC-dimension

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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