Abstract
Finding cohesive subgraphs in a network is a well-known problem in graph theory. Several alternative formulations of cohesive subgraph have been proposed, a notable example being s-club, which is a subgraph where each vertex is at distance at most s to the others. Here we consider the problem of covering a given graph with the minimum number of s-clubs. We study the computational and approximation complexity of this problem, when s is equal to 2 or 3. First, we show that deciding if there exists a cover of a graph with three 2-clubs is NP-complete, and that deciding if there exists a cover of a graph with two 3-clubs is NP-complete. Then, we consider the approximation complexity of covering a graph with the minimum number of 2-clubs and 3-clubs. We show that, given a graph \(G=(V,E)\) to be covered, covering G with the minimum number of 2-clubs is not approximable within factor \(O(|V|^{1/2 -\varepsilon })\), for any \(\varepsilon >0\), and covering G with the minimum number of 3-clubs is not approximable within factor \(O(|V|^{1 -\varepsilon })\), for any \(\varepsilon >0\). On the positive side, we give an approximation algorithm of factor \(2|V|^{1/2}\log ^{3/2} |V|\) for covering a graph with the minimum number of 2-clubs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alba, R.D.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3, 113–126 (1973)
Asahiro, Y., Doi, Y., Miyano, E., Samizo, K., Shimizu, H.: Optimal approximation algorithms for maximum distance-bounded subgraph problems. Algorithmica 80, 1834–1856 (2017)
Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)
Bourjolly, J., 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)
Cerioli, M.R., Faria, L., Ferreira, T.O., Martinhon, C.A.J., Protti, F., Reed, B.A.: Partition into cliques for cubic graphs: planar case, complexity and approximation. Discrete Appl. Math. 156(12), 2270–2278 (2008)
Cerioli, M.R., Faria, L., Ferreira, T.O., Protti, F.: A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation. RAIRO Theor. Inf. Appl. 45(3), 331–346 (2011)
Chang, M., Hung, L., Lin, C., Su, P.: Finding large k-clubs in undirected graphs. Computing 95(9), 739–758 (2013)
Desormeaux, W.J., Haynes, T.W., Henning, M.A., Yeo, A.: Total domination in graphs with diameter 2. J. Graph Theory 75(1), 91–103 (2014)
Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs. Graphs Comb. 27(3), 399–411 (2011)
Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Golovach, P.A., Heggernes, P., Kratsch, D., Rafiey, A.: Finding clubs in graph classes. Discrete Appl. Math. 174, 57–65 (2014)
Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. J. Graph Algorithms Appl. 19(1), 155–190 (2015)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Proceedings of a symposium on the Complexity of Computer Computations, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, 20–22 March 1972. The IBM Research Symposia Series, pp. 85–103. Plenum Press, New York (1972)
Komusiewicz, C.: Multivariate algorithmics for finding cohesive subnetworks. Algorithms 9(1), 21 (2016)
Komusiewicz, C., Sorge, M.: An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems. Discrete Appl. Math. 193, 145–161 (2015)
Laan, S., Marx, M., Mokken, R.J.: Close communities in social networks: boroughs and 2-clubs. Social Netw. Anal. Min. 6(1), 20:1–20:16 (2016)
Mokken, R.: Cliques, clubs and clans. Qual. Quant. Int. J. Methodol. 13(2), 161–173 (1979)
Mokken, R.J., Heemskerk, E.M., Laan, S.: Close communication and 2-clubs in corporate networks: Europe 2010. Social Netw. Anal. Min. 6(1), 40:1–40:19 (2016)
Pirwani, I.A., Salavatipour, M.R.: A weakly robust PTAS for minimum clique partition in unit disk graphs. Algorithmica 62(3–4), 1050–1072 (2012)
Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6(5), 883–891 (2012)
Zoppis, I., Dondi, R., Santoro, E., Castelnuovo, G., Sicurello, F., Mauri, G.: Optimizing social interaction - a computational approach to support patient engagement. In: Zwiggelaar, R., Gamboa, H., Fred, A.L.N., i Badia, S.B. (eds.) Proceedings of the 11th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2018) - Volume 5: HEALTHINF, Funchal, Madeira, Portugal, 19–21 January 2018, pp. 651–657. SciTePress (2018)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Dondi, R., Mauri, G., Sikora, F., Zoppis, I. (2018). Covering with Clubs: Complexity and Approximability. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-94667-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94666-5
Online ISBN: 978-3-319-94667-2
eBook Packages: Computer ScienceComputer Science (R0)