Abstract
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 \(d\ge 1\), 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 \(n^{1-\epsilon }\), for any \(\epsilon >0\). Moreover, it is known that, for any \(d\ge 2\), it is NP-hard to approximate MaxDBS within a factor \(n^{1/2-\epsilon }\), for any \(\epsilon >0\). 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.
Similar content being viewed by others
Notes
Note that for any \(\epsilon >0\), it could hold that the Euclidean distance between u and v is \(1+\epsilon \) but they are in different connected components of G, and hence d(u, v) is not necessarily bounded.
We assume that the diameter of G is greater than d, otherwise, G is a d-club.
References
Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom. 41(3), 365–375 (2009)
Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17(1), 1–9 (1997)
Alba, R.D.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3, 113–126 (1973)
Almeida, M.T., Carvalho, F.D.: Integer models and upper bounds for the \(3\)-club problem. Networks 60(3), 155–166 (2012)
Asahiro, Y., Doi, Y., Miyano, E., Samizo, K., Shimizu, H.: Optimal approximation algorithms for maximum distance-bounded subgraph problems. Algorithmica 80(6), 1834–1856 (2018)
Asahiro, Y., Miyano, E., Samizo, K.: Approximating maximum diameter-bounded subgraphs. In: LATIN 2010: Theoretical Informatics. Lecture Notes in Comput. Sci., vol. 6034, pp. 615–626. Springer, Berlin (2010)
Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)
Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Handbook of Combinatorial Optimization, Suppl. vol. A, pp. 1–74. Kluwer, Dordrecht (1999)
Bourjolly, J.-M., Laporte, G., Pesant, G.: An exact algorithm for the maximum \(k\)-club problem in an undirected graph. Eur. J. Oper. Res. 138(1), 21–28 (2002)
Carvalho, F.D., Almeida, M.T.: Upper bounds and heuristics for the \(2\)-club problem. Eur. J. Oper. Res. 210(3), 489–494 (2011)
Chang, M.-S., Hung, L.-J., Lin, C.-R., Su, P.-C.: Finding large \(k\)-clubs in undirected graphs. Computing 95(9), 739–758 (2013)
Chepoi, V., Estellon, B., Vaxès, Y.: Covering planar graphs with a fixed number of balls. Discrete Comput. Geom. 37(2), 237–244 (2007)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)
da Fonseca, G.D., de Figueiredo, C.M.H., Pereira de Sá, V.G., Machado, R.C.S.: Efficient sub-\(5\) approximations for minimum dominating sets in unit disk graphs. Theor. Comput. Sci. 540/541, 70–81 (2014)
Gräf, A., Stumpf, M., Weißenfels, G.: On coloring unit disk graphs. Algorithmica 20(3), 277–293 (1998)
Har-Peled, S.: Geometric Approximation Algorithms. Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence (2011)
Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)
Matoušek, J.: Bounded VC-dimension implies a fractional Helly theorem. Discrete Comput. Geom. 31(2), 251–255 (2004)
Pattillo, J., Wang, Y., Butenko, S.: Approximating \(2\)-cliques in unit disk graphs. Discrete Appl. Math. 166, 178–187 (2014)
Pattillo, J., Youssef, N., Butenko, S.: On clique relaxation models in network analysis. Eur. J. Oper. Res. 226(1), 9–18 (2013)
Veremyev, A., Boginski, V.: Identifying large robust network clusters via new compact formulations of maximum \(k\)-club problems. Eur. J. Oper. Res. 218(2), 316–326 (2012)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3, 103–128 (2007)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: János Pach
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abu-Affash, A.K., Carmi, P., Maheshwari, A. et al. Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs. Discrete Comput Geom 66, 1401–1414 (2021). https://doi.org/10.1007/s00454-021-00327-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-021-00327-y
Keywords
- Approximation algorithms
- Maximum diameter-bounded subgraph
- Unit disk graphs
- Fractional Helly theorem
- VC-dimension