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A Spectral Algorithm for Finding Maximum Cliques in Dense Random Intersection Graphs

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SOFSEM 2023: Theory and Practice of Computer Science (SOFSEM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13878))

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Abstract

In a random intersection graph \(G_{n,m,p}\), each of n vertices selects a random subset of a set of m labels by including each label independently with probability p and edges are drawn between vertices that have at least one label in common. Among other applications, such graphs have been used to model social networks, in which individuals correspond to vertices and various features (e.g. ideas, interests) correspond to labels; individuals sharing at least one common feature are connected and this is abstracted by edges in random intersection graphs. In this paper, we consider the problem of finding maximum cliques when the input graph is \(G_{n,m,p}\). Current algorithms for this problem are successful with high probability only for relatively sparse instances, leaving the dense case mostly unexplored. We present a spectral algorithm for finding large cliques that processes vertices according to respective values in the second largest eigenvector of the adjacency matrix of induced subgraphs of the input graph corresponding to common neighbors of small cliques. Leveraging on the Single Label Clique Theorem from [16], we were able to construct random instances, without the need to externally plant a large clique in the input graph. In particular, we used label choices to determine the maximum clique and then concealed label information by just giving the adjacency matrix of \(G_{n, m, p}\) as input to the algorithm. Our experimental evaluation showed that our spectral algorithm clearly outperforms existing polynomial time algorithms, both with respect to the failure probability and the approximation guarantee metrics, especially in the dense regime, thus suggesting that spectral properties of random intersection graphs may be also used to construct efficient algorithms for other NP-hard graph theoretical problems as well.

Christoforos Raptopoulos was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project Number: 704).

Paul Spirakis was supported by the NeST initiative of the EEE and CS of the University of Liverpool and by the EPSRC grant EP/P02002X/1.

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References

  1. Alon, N., Krivelevich, M., Sudakov, B.: Finding a large hidden clique in a random graph. Random Struct. Algor. 13, 457–466 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Behrisch, M., Taraz, A.: Efficiently covering complex networks with cliques of similar vertices. Theor. Comput. Sci. 355(1), 37–47 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bloznelis, M., Godehardt, E., Jaworski, J., Kurauskas, V., Rybarczyk, K.: Recent progress in complex network analysis: properties of random intersection graphs. In: Lausen, B., Krolak-Schwerdt, S., Böhmer, M. (eds.) Data Science, Learning by Latent Structures, and Knowledge Discovery. SCDAKO, pp. 79–88. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-44983-7_7

    Chapter  Google Scholar 

  4. Bloznelis, M., Godehardt, E., Jaworski, J., Kurauskas, V., Rybarczyk, K.: Recent progress in complex network analysis: models of random intersection graphs. In: Lausen, B., Krolak-Schwerdt, S., Böhmer, M. (eds.) Data Science, Learning by Latent Structures, and Knowledge Discovery. SCDAKO, pp. 69–78. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-44983-7_6

    Chapter  Google Scholar 

  5. Bloznelis, M., Kurauskas, V.: Large cliques in sparse random intersection graphs. Electr. J. Comb. 24(2), P2.5 (2017)

    Google Scholar 

  6. Christodoulou, F., Nikoletseas, S., Raptopoulos, C., Spirakis, P.: A spectral algorithm for finding maximum cliques in dense random intersection graphs (2022). https://doi.org/10.48550/ARXIV.2210.02121, https://arxiv.org/abs/2210.02121

  7. Fill, J.A., Sheinerman, E.R., Singer-Cohen, K.B.: Random intersection graphs when \(m = \omega (n)\): an equivalence theorem relating the evolution of the \(g(n, m, p)\) and \(g(n, p)\) models. Random Struct. Algor. 16(2), 156–176 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Friedrich, T., Hercher, C.: On the kernel size of clique cover reductions for random intersection graphs. J. Discrete Algorithms 34, 128–136 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grimmett, G.R., McDiarmid, C.: On coloring random graphs. Math. Proc. Cambridge Philos. Soc. 77, 313–324 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Håstad, J.: Clique is hard to approximate within \(n^{1-\varepsilon }\). Acta Math. 182, 105–142 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jianer, C., Xiuzhen, H., Iyad, A.K., Ge, X.: Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci. 72(8), 1346–1367 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karoński, M., Scheinerman, E.R., Singer-Cohen, K.B.: On random intersection graphs: the subgraph problem. Comb. Probab. Comput. 8, 131–159 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of computer computations, pp. 85–103. Plenum Press (1972)

    Google Scholar 

  14. Karp, R.M.: Probabilistic analysis of some combinatorial search problems. In: Algorithms and Complexity: New Directions and Recent Results, pp. 85–103. Academic Press (1976)

    Google Scholar 

  15. Nikoletseas, S.E., Raptopoulos, C.L., Spirakis, P.G.: Communication and security in random intersection graphs models. In: 12th IEEE International Symposium on a World of Wireless, Mobile and Multimedia Networks (WOWMOM), pp. 1–6 (2011)

    Google Scholar 

  16. Nikoletseas, S.E., Raptopoulos, C.L., Spirakis, P.G.: Maximum cliques in graphs with small intersection number and random intersection graphs. In: Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 728–739 (2012)

    Google Scholar 

  17. Nikoletseas, S.E., Raptopoulos, C.L., Spirakis, P.G.: Maximum cliques in graphs with small intersection number and random intersection graphs. Comput. Sci. Rev. 39, 100353 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rybarczyk, K.: Equivalence of a random intersection graph and \(g(n, p)\). Random Struct. Algor. 38(1–2), 205–234 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Singer-Cohen, K.B.: Random intersection graphs. Ph.D. thesis, John Hopkins University (1995)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Gran Sasso Science Institute, L’Aquila, Italy

    Filippos Christodoulou

  2. Computer Engineering and Informatics Department, University of Patras, Patras, Greece

    Sotiris Nikoletseas & Paul G. Spirakis

  3. Computer Technology Institute & Press Diophantus (CTI), Patras, Greece

    Sotiris Nikoletseas & Christoforos Raptopoulos

  4. Department of Computer Science, University of Liverpool, Liverpool, UK

    Paul G. Spirakis

Authors
  1. Filippos Christodoulou
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  2. Sotiris Nikoletseas
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  3. Christoforos Raptopoulos
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  4. Paul G. Spirakis
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Corresponding author

Correspondence to Filippos Christodoulou .

Editor information

Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Leszek Gąsieniec

7 Appendix

7 Appendix

1.1 7.1 Greedy-Clique Algorithm

The pseudocode of the GREEDY-CLIQUE Algorithm from [5] is shown below.

figure b

1.2 7.2 Mono-Clique Algorithm

The pseudocode of the MONO-CLIQUE Algorithm from [5] is shown below.

figure c

1.3 7.3 Maximum-Clique Algorithm

The pseudocode of the Maximum-Clique Algorithm from [2] is shown below.

figure d

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Christodoulou, F., Nikoletseas, S., Raptopoulos, C., Spirakis, P.G. (2023). A Spectral Algorithm for Finding Maximum Cliques in Dense Random Intersection Graphs. In: Gąsieniec, L. (eds) SOFSEM 2023: Theory and Practice of Computer Science. SOFSEM 2023. Lecture Notes in Computer Science, vol 13878. Springer, Cham. https://doi.org/10.1007/978-3-031-23101-8_2

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  • DOI: https://doi.org/10.1007/978-3-031-23101-8_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-23100-1

  • Online ISBN: 978-3-031-23101-8

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