Document Open Access Logo

Planted Models for the Densest k-Subgraph Problem

Authors Yash Khanna, Anand Louis

PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2020.27.pdf
  • Filesize: 0.63 MB
  • 18 pages

Document Identifiers

Author Details

Yash Khanna
  • Indian Institute of Science, Bangalore, India
Anand Louis
  • Indian Institute of Science, Bangalore, India

Funding

  • Louis, Anand: AL was supported in part by SERB Award ECR/2017/003296 and a Pratiksha Trust Young Investigator Award.

Acknowledgements

We thank Rakesh Venkat for helpful discussions. We also thank the anonymous reviewers for their suggestions and comments on earlier versions of this paper.

Cite AsGet BibTex

Yash Khanna and Anand Louis. Planted Models for the Densest k-Subgraph Problem. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FSTTCS.2020.27

Abstract

Given an undirected graph G, the Densest k-subgraph problem (DkS) asks to compute a set S ⊂ V of cardinality |S| ≤ k such that the weight of edges inside S is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a 𝒪(n^{1/4 + ε}) approximation in time n^{𝒪(1/ε)}, for any ε > 0. We ask what are some "easier" instances of this problem? We propose some natural semi-random models of instances with a planted dense subgraph, and study approximation algorithms for computing the densest subgraph in them. These models are inspired by the semi-random models of instances studied for various other graph problems such as the independent set problem, graph partitioning problems etc. For a large range of parameters of these models, we get significantly better approximation factors for the Densest k-subgraph problem. Moreover, our algorithm recovers a large part of the planted solution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Semidefinite programming
  • Theory of computation → Discrete optimization
  • Theory of computation → Graph algorithms analysis
Keywords
  • Densest k-Subgraph
  • Semi-Random models
  • Planted Models
  • Semidefinite Programming
  • Approximation Algorithms
  • Beyond Worst Case Analysis

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM J. Comput., 26(6):1733-1748, December 1997. URL: https://doi.org/10.1137/S0097539794270248.
  2. Brendan P. Ames. Guaranteed recovery of planted cliques and dense subgraphs by convex relaxation. J. Optim. Theory Appl., 167(2):653–675, November 2015. URL: https://doi.org/10.1007/s10957-015-0777-x.
  3. Yuichi Asahiro, Kazuo Iwama, Hisao Tamaki, and Takeshi Tokuyama. Greedily finding a dense subgraph. In Algorithm Theory - SWAT'96, pages 136-148, Berlin, Heidelberg, 1996. Springer Berlin Heidelberg. Google Scholar
  4. Nikhil Bansal, Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph Naor, and Roy Schwartz. Min-max graph partitioning and small set expansion. SIAM J. Comput., 43(2):872-904, 2014. URL: https://doi.org/10.1137/120873996.
  5. Aditya Bhaskara. Finding dense structures in graphs and matrices. PhD thesis, Princeton University, 2012. URL: https://www.cs.utah.edu/~bhaskara/files/thesis.pdf.
  6. Aditya Bhaskara, Moses Charikar, Eden Chlamtac, Uriel Feige, and Aravindan Vijayaraghavan. 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, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 201-210, 2010. URL: https://doi.org/10.1145/1806689.1806719.
  7. Aditya Bhaskara, Moses Charikar, Aravindan Vijayaraghavan, Venkatesan Guruswami, and Yuan Zhou. Polynomial integrality gaps for strong sdp relaxations of densest k-subgraph. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’12, page 388–405, USA, 2012. Society for Industrial and Applied Mathematics. Google Scholar
  8. Polina Bombina and Brendan Ames. Convex optimization for the densest subgraph and densest submatrix problems, 2019. URL: http://arxiv.org/abs/1904.03272.
  9. Mark Braverman, Young Kun Ko, Aviad Rubinstein, and Omri Weinstein. Eth hardness for densest-k-subgraph with perfect completeness. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 1326-1341, Philadelphia, PA, USA, 2017. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=3039686.3039772.
  10. Moses Charikar. Greedy approximation algorithms for finding dense components in a graph. In Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX '00, pages 84-95, Berlin, Heidelberg, 2000. Springer-Verlag. URL: http://dl.acm.org/citation.cfm?id=646688.702972.
  11. Amin Coja-Oghlan. Colouring semirandom graphs. Comb. Probab. Comput., 16(4):515-552, July 2007. URL: https://doi.org/10.1017/S0963548306007917.
  12. Roee David and Uriel Feige. On the effect of randomness on planted 3-coloring models. In Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, STOC '16, pages 77-90, New York, NY, USA, 2016. ACM. URL: https://doi.org/10.1145/2897518.2897561.
  13. Uriel Feige and Joe Kilian. Heuristics for semirandom graph problems. J. Comput. Syst. Sci., 63(4):639–671, December 2001. URL: https://doi.org/10.1006/jcss.2001.1773.
  14. Uriel Feige, Guy Kortsarz, and David Peleg. The dense k-subgraph problem. Algorithmica, 29(3):410-421, 2001. URL: https://doi.org/10.1007/s004530010050.
  15. Uriel Feige and Michael Langberg. Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithms, 41(2):174-211, 2001. URL: https://doi.org/10.1006/jagm.2001.1183.
  16. Uriel Feige and Michael Seltser. On the densest k-subgraph problem. Algorithmica, 29:2001, 1997. Google Scholar
  17. G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. Comput., 18(1):30-55, February 1989. URL: https://doi.org/10.1137/0218003.
  18. A. V. Goldberg. Finding a maximum density subgraph. Technical report, University of California at Berkeley, Berkeley, CA, USA, 1984. Google Scholar
  19. B. Hajek, Y. Wu, and J. Xu. Achieving exact cluster recovery threshold via semidefinite programming: Extensions. IEEE Transactions on Information Theory, 62(10):5918-5937, 2016. Google Scholar
  20. Bruce Hajek, Yihong Wu, and Jiaming Xu. Computational Lower Bounds for Community Detection on Random Graphs. arXiv e-prints, page arXiv:1406.6625, June 2014. URL: http://arxiv.org/abs/1406.6625.
  21. Bruce Hajek, Yihong Wu, and Jiaming Xu. Semidefinite programs for exact recovery of a hidden community. Journal of Machine Learning Research, 49(June):1051-1095, June 2016. 29th Conference on Learning Theory, COLT 2016 ; Conference date: 23-06-2016 Through 26-06-2016. Google Scholar
  22. Ravi Kannan and V Vinay. Analyzing the structure of large graphs. Rheinische Friedrich-Wilhelms-Universität Bonn Bonn, 1999. Google Scholar
  23. Subhash Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput., 36(4):1025-1071, December 2006. URL: https://doi.org/10.1137/S0097539705447037.
  24. Samir Khuller and Barna Saha. On finding dense subgraphs. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I, ICALP '09, pages 597-608, Berlin, Heidelberg, 2009. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-02927-1_50.
  25. Alexandra Kolla, Konstantin Makarychev, and Yury Makarychev. How to play unique games against a semi-random adversary: Study of semi-random models of unique games. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 443-452, 2011. URL: https://doi.org/10.1109/FOCS.2011.78.
  26. Anand Louis and Rakesh Venkat. Semi-random graphs with planted sparse vertex cuts: Algorithms for exact and approximate recovery. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 101:1-101:15, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.101.
  27. Anand Louis and Rakesh Venkat. Planted models for k-way edge and vertex expansion. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2019, December 11-13, 2019, Bombay, India, pages 23:1-23:15, 2019. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2019.23.
  28. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC '12, pages 367-384, New York, NY, USA, 2012. ACM. URL: https://doi.org/10.1145/2213977.2214013.
  29. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Constant factor approximation for balanced cut in the pie model. In Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 41-49, New York, NY, USA, 2014. ACM. URL: https://doi.org/10.1145/2591796.2591841.
  30. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Learning communities in the presence of errors. In Vitaly Feldman, Alexander Rakhlin, and Ohad Shamir, editors, 29th Annual Conference on Learning Theory, volume 49 of Proceedings of Machine Learning Research, pages 1258-1291, Columbia University, New York, New York, USA, 2016. PMLR. URL: http://proceedings.mlr.press/v49/makarychev16.html.
  31. Pasin Manurangsi. Almost-polynomial ratio eth-hardness of approximating densest k-subgraph. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 954-961, 2017. URL: https://doi.org/10.1145/3055399.3055412.
  32. Theo McKenzie, Hermish Mehta, and Luca Trevisan. A new algorithm for the robust semi-random independent set problem. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 738-746, 2020. URL: https://doi.org/10.1137/1.9781611975994.45.
  33. F. McSherry. Spectral partitioning of random graphs. In Proceedings 42nd IEEE Symposium on Foundations of Computer Science, pages 529-537, 2001. Google Scholar
  34. Andrea Montanari. Finding one community in a sparse graph. Journal of Statistical Physics, 161, February 2015. URL: https://doi.org/10.1007/s10955-015-1338-2.
  35. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC '10, pages 755-764, New York, NY, USA, 2010. ACM. URL: https://doi.org/10.1145/1806689.1806792.
  36. Prasad Raghavendra, David Steurer, and Prasad Tetali. Approximations for the isoperimetric and spectral profile of graphs and related parameters. In Proceedings of the Forty-Second ACM Symposium on Theory of Computing, STOC ’10, page 631–640, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1806689.1806776.
  37. Anand Srivastav and Katja Wolf. Finding dense subgraphs with semidefinite programming. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX '98, pages 181-191, London, UK, UK, 1998. Springer-Verlag. URL: http://dl.acm.org/citation.cfm?id=646687.702946.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact