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Detecting and Characterizing Small Dense Bipartite-Like Subgraphs by the Bipartiteness Ratio Measure

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Algorithms and Computation (ISAAC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8283))

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Abstract

We study the problem of finding and characterizing subgraphs with small bipartiteness ratio. We give a bicriteria approximation algorithm SwpDB such that if there exists a subset S of volume at most k and bipartiteness ratio θ, then for any 0 < ε < 1/2, it finds a set S′ of volume at most 2k 1 + ε and bipartiteness ratio at most \(4\sqrt{\theta/\epsilon}\). By combining a truncation operation, we give a local algorithm LocDB, which has asymptotically the same approximation guarantee as the algorithm SwpDB on both the volume and bipartiteness ratio of the output set, and runs in time O(ε 2 θ − 2 k 1 + εln 3 k), independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the kth largest eigenvalue of the Laplacian of the graph.

The research is partially supported by NSFC distinguished young investigator award number 60325206, and its matching fund from the Hundred-Talent Program of the Chinese Academy of Sciences. Both authors are partially supported by the Grand Project “Network Algorithms and Digital Information” of the Institute of software, Chinese Academy of Sciences. The second author acknowledges the support of ERC grant No. 307696 and NSFC 61003030.

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References

  1. Alon, N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Milman, V.: λ 1, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B 38(1), 73–88 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersen, R.: A local algorithm for finding dense subgraphs. ACM Transactions on Algorithms (TALG) 6(4), 60 (2010)

    Google Scholar 

  4. Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: Avrachenkov, K., Donato, D., Litvak, N. (eds.) WAW 2009. LNCS, vol. 5427, pp. 25–37. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  5. Andersen, R., Chung, F., Lang, K.: Local graph partitioning using pagerank vectors. In: IEEE Symposium on Foundations of Computer Science (2006)

    Google Scholar 

  6. Andersen, R., Peres, Y.: Finding sparse cuts locally using evolving sets. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (2009)

    Google Scholar 

  7. Arora, S., Barak, B., Steurer, D.: Subexponential algorithms for unique games and related problems. In: Foundations of Computer Science, FOCS (2010)

    Google Scholar 

  8. Arora, S., Ge, R., Sachdeva, S., Schoenebeck, G.: Finding overlapping communities in social networks: toward a rigorous approach. In: Proceedings of the 13th ACM Conference on Electronic Commerce (2012)

    Google Scholar 

  9. Balcan, M.F., Borgs, C., Braverman, M., Chayes, J., Teng, S.: Finding endogenously formed communities. In: SODA (2013)

    Google Scholar 

  10. Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an o(n 1/4) approximation for densest k-subgraph. In: Proceedings of the 42nd ACM Symposium on Theory of Computing (2010)

    Google Scholar 

  11. Charikar, M.: Greedy approximation algorithms for finding dense components in a graph. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 84–95. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  12. Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kannan, R., Vempala, S., Vetta, A.: On clusterings: Good, bad and spectral. J. ACM 51(3), 497–515 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khuller, S., Saha, B.: On finding dense subgraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 597–608. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  15. Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the web for emerging cyber-communities. In: Proceedings of the 8th International Conference on World Wide Web, pp. 1481–1493 (1999)

    Google Scholar 

  16. Kwok, T.C., Lau, L.C.: Finding small sparse cuts by random walk. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM 2012. LNCS, vol. 7408, pp. 615–626. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  17. Lee, J., Oveis Gharan, S., Trevisan, L.: Multi-way spectral partitioning and higher-order cheeger inequalities. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (2012)

    Google Scholar 

  18. Leskovec, J., Lang, K., Dasgupta, A., Mahoney, M.: Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics 6(1), 29–123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leskovec, J., Lang, K.J., Mahoney, M.: Empirical comparison of algorithms for network community detection. In: Proceedings of the 19th International Conference on World Wide Web, pp. 631–640 (2010)

    Google Scholar 

  20. Liao, C.S., Lu, K., Baym, M., Singh, R., Berger, B.: IsoRankN: spectral methods for global alignment of multiple protein networks. Bioinformatics, i253–i258 (2009)

    Google Scholar 

  21. Louis, A., Raghavendra, P., Tetali, P., Vempala, S.: Many sparse cuts via higher eigenvalues. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (2012)

    Google Scholar 

  22. Lovász, L., Simonovits, M.: The mixing rate of markov chains, an isoperimetric inequality, and computing the volume. In: Foundations of Computer Science (FOCS), pp. 346–354 (1990)

    Google Scholar 

  23. Lovász, L., Simonovits, M.: Random walks in a convex body and an improved volume algorithm. Random structures & Algorithms 4(4), 359–412 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. O’Donnell, R., Witmer, D.: Improved small-set expansion from higher eigenvalues. Arxiv preprint arXiv:1204.4688 (2012)

    Google Scholar 

  25. Oveis Gharan, S., Trevisan, L.: Approximating the expansion profile and almost optimal local graph clustering. In: 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS (2012)

    Google Scholar 

  26. Peng, P.: A local algorithm for finding dense bipartite-like subgraphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 145–156. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  27. Raghavendra, P., Steurer, D.: Graph expansion and the unique games conjecture. In: 42nd ACM Symposium on Theory of Computing, pp. 755–764 (2010)

    Google Scholar 

  28. Sinclair, A., Jerrum, M.: Approximate counting, uniform generation and rapidly mixing markov chains. Information and Computation 82(1), 93–133 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Spielman, D., Teng, S.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, pp. 81–90 (2004)

    Google Scholar 

  30. Spielman, D., Teng, S.: A local clustering algorithm for massive graphs and its application to nearly-linear time graph partitioning. arXiv:0809.3232 (2008)

    Google Scholar 

  31. Spielman, D.A.: Algorithms, graph theory, and linear equations. In: Proceedings of the International Congress of Mathematicians IV, pp. 2698–2722 (2010)

    Google Scholar 

  32. Teng, S.-H.: The laplacian paradigm: Emerging algorithms for massive graphs. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) TAMC 2010. LNCS, vol. 6108, pp. 2–14. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  33. Trevisan, L.: Max cut and the smallest eigenvalue. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 263–272 (2009)

    Google Scholar 

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Author information

Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, China

    Angsheng Li & Pan Peng

  2. Department of Computer Science, Technische Universität Dortmund, Germany

    Pan Peng

Authors
  1. Angsheng Li
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  2. Pan Peng
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, 100084, Hong Kong, China

    Leizhen Cai

  2. Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, 100044, Hong Kong, China

    Siu-Wing Cheng

  3. Department of Computer Science, University of Hong Kong, Porfkulam, 100084, Hong Kong, China

    Tak-Wah Lam

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Li, A., Peng, P. (2013). Detecting and Characterizing Small Dense Bipartite-Like Subgraphs by the Bipartiteness Ratio Measure. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_61

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  • DOI: https://doi.org/10.1007/978-3-642-45030-3_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45029-7

  • Online ISBN: 978-3-642-45030-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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