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 - 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 - 10.4230/LIPIcs.SoCG.2018.2
DO - 10.4230/LIPIcs.SoCG.2018.2
M3 - Conference contribution
AN - SCOPUS:85048989349
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 -