TY - GEN

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: Work was supported by Grant 2016116 from the United States - Israel Binational Science Foundation. Work was supported by Grant 2016116 from the United States - Israel Binational Science Foundation. Work was supported by NSERC Work was supported by NSERC Work was supported by NSERC Work was partially supported by Grant 635/16 from the Israel Science Foundation Funding Information: Work was supported by Grant 2016116 from the United States – Israel Binational Science Foundation. 2 Work was supported by Grant 2016116 from the United States – Israel Binational Science Foundation. 3 Work was supported by NSERC 4 Work was supported by NSERC 5 Work was supported by NSERC 6 Work was partially supported by Grant 635/16 from the Israel Science Foundation Publisher Copyright: © A.Karim Abu-Affash, Paz Carmi, Anil Maheshwari, Pat Morin, Michiel Smid, and Shakhar Smorodinsky; licensed under Creative Commons License CC-BY 34th Symposium on Computational Geometry (SoCG 2018).

PY - 2018/6/1

Y1 - 2018/6/1

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 an integer 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 an integer 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=85048989349&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.SoCG.2018.2

DO - https://doi.org/10.4230/LIPIcs.SoCG.2018.2

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 21

EP - 212

BT - 34th International Symposium on Computational Geometry, SoCG 2018

A2 - Toth, Csaba D.

A2 - Speckmann, Bettina

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 34th International Symposium on Computational Geometry, SoCG 2018

Y2 - 11 June 2018 through 14 June 2018

ER -