Abstract
The Edge Clique Cover (ECC) problem is concerned with covering edges of a graph with the minimum number of cliques, which is an NP-hard problem. This problem has many real-life applications, such as, in computational biology, food science, efficient representation of pairwise information, and so on. In this work we propose using a compact representation of network data based on sparse matrix data structures. Building upon an existing ECC heuristic due to Kellerman we proffer adding vertices during the clique-growing step of the algorithm in judiciously chosen degree-based orders. On a set of standard benchmark instances our ordered approach produced smaller sized clique cover compared to unordered processing.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Kepner, J., Gilbert, J.: Graph Algorithms in the Language of Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2011)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Data reduction and exact algorithms for clique cover. J. Exp. Algorithmics (JEA). 13, 2–15 (2009)
Hossain, S., Khan, A.I.: Exact coloring of sparse matrices. In: Kilgour, D.M., et al. (eds.) Recent Advances in Mathematical and Statistical Methods. Springer Proceedings in Mathematics and Statistics, vol. 259, pp. 23–36. Springer Nature, Switzerland AG (2018)
Kepner, J., Jananthan, H.: Mathematics of Big Data: Spreadsheets, Databases, Matrices, and Graphs. MIT Press (2018)
Hasan, M., Hossain, S., Khan, A.I., Mithila, N.H., Suny, A.H.: DSJM: a software toolkit for direct determination of sparse Jacobian matrices. In: Greuel, G.M., Koch, T., Paule, P., Sommese, A. (eds.) ICMS2016, pp. 425–434. Springer International Publishing, Switzerland (2016)
Hossain, S., Suny, A.H.: Determination of large sparse derivative matrices: structural: orthogonality and structural degeneracy. In: Randerath, B., Röglin, H., Peis, B., Schaudt, O., Schrader, R., Vallentin, F., Weil, V. (eds.) 15th Cologne-Twente Workshop on Graphs & Combinatorial Optimizationpp, pp. 83–87. Cologne, Germany (2017)
Kou, L.T., Stockmeyer, L.J., Wong, C.K.: Covering edges by cliques with regard to keyword conflicts and intersection graphs. Commun. ACM. 21(2), 135–139 (1978)
Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press (1994)
James, O.: Contentment in graph theory: covering graphs with cliques. Indagationes Mathematicae (Proceedings), vol. 80, no. 5. North-Holland (1977)
Gramm, J., Guo, J., Hüffner, F. Niedermeier, R.: Data reduction, exact and heuristic algorithms for clique cover. In: Proceedings of the Eighth Workshop on Algorithm Engineering and Experiments (ALENEX). pp. 86–94. SIAM (2006)
Kellerman, E.: Determination of keyword conflict. IBM Tech. Discl. Bull. 16(2), 544–546 (1973)
SuiteSparse Matrix Collection. https://sparse.tamu.edu/. Accessed 02 Oct 2019
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R., Piepho, H., Schmid, R.: Algorithms for compact letter displays: comparison and evaluation. Comput. Stat. Data Anal. 52, 725–736 (2007)
Nestrud, M.A., Ennis, J.M., Fayle, C.M., Ennis, D.M., Lawless, H.T.: Validating a graph theoretic screening approach to food item combinations. J. Sens. Stud. 26(5), 331–338 (2011)
Blanchette, M., Kim, E., Vetta, A.: Clique cover on sparse networks. In: 2012 Proceedings of the Fourteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 93–102. Society for Industrial and Applied Mathematics, 2012 Jan 16
Ennis, J.M., Ennis, D.M.: Efficient representation of pairwise sensory information. IFPress 15(3), 3–4 (2012)
Tinhofer, G.: Generating graphs uniformly at random. In: Tinhofer, G., Mayr, E., Noltemeier, H., Syslo, M.M. (eds.) Computational Graph Theory. Computing Supplementum, vol. 7, pp. 235–255. Springer, Vienna (1990)
Hossain, S., Steihaug, T.: Graph models and their efficient implementation for sparse Jacobian matrix determination. Discrete Appl. Math. 161(12), 1747–1754 (2013)
Hossain, S., Steihaug, T.: Optimal direct determination of sparse Jacobian matrices. Optim. Methods Softw. (2012). https://doi.org/10.1080/10556788.2012.693927
Acknowledgements
We thank referees for their many valuable suggestions that helped improve the paper. This research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) under Discovery Grants Program.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Abdullah, W.M., Hossain, S., Khan, M.A. (2021). Covering Large Complex Networks by Cliques—A Sparse Matrix Approach. In: Kilgour, D.M., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Developments in Mathematical, Statistical and Computational Sciences. AMMCS 2019. Springer Proceedings in Mathematics & Statistics, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-030-63591-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-63591-6_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-63590-9
Online ISBN: 978-3-030-63591-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)