TY - JOUR

T1 - Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

AU - Abu-Affash, A. Karim

AU - Carmi, Paz

AU - Maheshwari, Anil

AU - Morin, Pat

AU - Smid, Michiel

AU - Smorodinsky, Shakhar

N1 - Funding Information:
The authors would like to thank the Fields Institute for hosting the workshop in Ottawa and for their financial support. The authors would also like to thank the anonymous reviewers for their helpful comments and suggestions.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/12/1

Y1 - 2021/12/1

N2 - 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.

AB - 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.

KW - Approximation algorithms

KW - Fractional Helly theorem

KW - Maximum diameter-bounded subgraph

KW - Unit disk graphs

KW - VC-dimension

UR - http://www.scopus.com/inward/record.url?scp=85113668441&partnerID=8YFLogxK

U2 - 10.1007/s00454-021-00327-y

DO - 10.1007/s00454-021-00327-y

M3 - Article

AN - SCOPUS:85113668441

SN - 0179-5376

VL - 66

SP - 1401

EP - 1414

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -