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Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

Authors R. Krithika , V. K. Kutty Malu, Roohani Sharma, Prafullkumar Tale

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Author Details

R. Krithika
  • Indian Institute of Technology Palakkad, India
V. K. Kutty Malu
  • Indian Institute of Technology Palakkad, India
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Prafullkumar Tale
  • Indian Institute of Science Education and Research Pune, India

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R. Krithika, V. K. Kutty Malu, Roohani Sharma, and Prafullkumar Tale. Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 8:1-8:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


A bipartite graph is called a biclique if it is a complete bipartite graph and a biclique is called a balanced biclique if it has equal number of vertices in both parts of its bipartition. In this work, we initiate the complexity study of Biclique Contraction and Balanced Biclique Contraction. In these problems, given as input a graph G and an integer k, the objective is to determine whether one can contract at most k edges in G to obtain a biclique and a balanced biclique, respectively. We first prove that these problems are NP-complete even when the input graph is bipartite. Next, we study the parameterized complexity of these problems and show that they admit single exponential-time FPT algorithms when parameterized by the number k of edge contractions. Then, we show that Balanced Biclique Contraction admits a quadratic vertex kernel while Biclique Contraction does not admit any polynomial compression (or kernel) unless NP ⊆ coNP/poly.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • contraction
  • bicliques
  • balanced bicliques
  • parameterized complexity


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  1. Takao Asano and Tomio Hirata. Edge-contraction problems. J. Comput. Syst. Sci., 26(2):197-208, 1983. URL:
  2. Ivan Bliznets, Fedor V. Fomin, Marcin Pilipczuk, and Michal Pilipczuk. A subexponential parameterized algorithm for proper interval completion. SIAM J. Discret. Math., 29(4):1961-1987, 2015. URL:
  3. Ivan Bliznets, Fedor V. Fomin, Marcin Pilipczuk, and Michal Pilipczuk. Subexponential parameterized algorithm for interval completion. ACM Trans. Algorithms, 14(3):35:1-35:62, 2018. URL:
  4. Andries E. Brouwer and Henk Jan Veldman. Contractibility and np-completeness. J. Graph Theory, 11(1):71-79, 1987. URL:
  5. Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett., 58(4):171-176, 1996. URL:
  6. Leizhen Cai and Chengwei Guo. Contracting few edges to remove forbidden induced subgraphs. In Proceedings of the 8th International Symposium on Parameterized and Exact Computation, pp. 97-109, 2013. Google Scholar
  7. Yixin Cao. Linear recognition of almost interval graphs. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1096-1115. SIAM, 2016. URL:
  8. Yixin Cao. Unit interval editing is fixed-parameter tractable. Inf. Comput., 253:109-126, 2017. URL:
  9. Yixin Cao and Dániel Marx. Interval deletion is fixed-parameter tractable. ACM Trans. Algorithms, 11(3):21:1-21:35, 2015. URL:
  10. Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. Algorithmica, 75(1):118-137, 2016. URL:
  11. Dipayan Chakraborty and R. B. Sandeep. Contracting edges to destroy a pattern: A complexity study. In Henning Fernau and Klaus Jansen, editors, Fundamentals of Computation Theory - 24th International Symposium, FCT 2023, Trier, Germany, September 18-21, 2023, Proceedings, volume 14292 of Lecture Notes in Computer Science, pages 118-131. Springer, 2023. URL:
  12. Christophe Crespelle, Pål Grønås Drange, Fedor V. Fomin, and Petr A. Golovach. A survey of parameterized algorithms and the complexity of edge modification. CoRR, abs/2001.06867, 2020. Google Scholar
  13. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  14. Konrad K. Dabrowski and Daniël Paulusma. Contracting bipartite graphs to paths and cycles. Inf. Process. Lett., 127:37-42, 2017. URL:
  15. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and ids. ACM Trans. Algorithms, 11(2):13:1-13:20, 2014. URL:
  16. Pål Grønås Drange, Markus Fanebust Dregi, Daniel Lokshtanov, and Blair D. Sullivan. On the threshold of intractability. J. Comput. Syst. Sci., 124:1-25, 2022. URL:
  17. Pål Grønås Drange, Fedor V. Fomin, Michal Pilipczuk, and Yngve Villanger. Exploring subexponential parameterized complexity of completion problems. In Ernst W. Mayr and Natacha Portier, editors, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, March 5-8, 2014, Lyon, France, volume 25 of LIPIcs, pages 288-299. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2014. URL:
  18. Pål Grønås Drange and Michal Pilipczuk. A polynomial kernel for trivially perfect editing. Algorithmica, 80(12):3481-3524, 2018. URL:
  19. Jirí Fiala, Marcin Kaminski, and Daniël Paulusma. A note on contracting claw-free graphs. Discret. Math. Theor. Comput. Sci., 15(2):223-232, 2013. URL:
  20. Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Yngve Villanger. Tight bounds for parameterized complexity of cluster editing with a small number of clusters. J. Comput. Syst. Sci., 80(7):1430-1447, 2014. URL:
  21. Fedor V. Fomin, Daniel Lokshtanov, Ivan Mihajlin, Saket Saurabh, and Meirav Zehavi. Computation of hadwiger number and related contraction problems: Tight lower bounds. ACM Trans. Comput. Theory, 13(2):10:1-10:25, 2021. URL:
  22. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. Google Scholar
  23. Fedor V. Fomin and Yngve Villanger. Subexponential parameterized algorithm for minimum fill-in. SIAM J. Comput., 42(6):2197-2216, 2013. URL:
  24. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  25. Petr A. Golovach, Pim van 't Hof, and Daniël Paulusma. Obtaining planarity by contracting few edges. Theor. Comput. Sci., 476:38-46, 2013. URL:
  26. Sylvain Guillemot and Dániel Marx. A faster FPT algorithm for bipartite contraction. Inf. Process. Lett., 113(22-24):906-912, 2013. URL:
  27. Pinar Heggernes, Pim van 't Hof, Benjamin Lévêque, Daniel Lokshtanov, and Christophe Paul. Contracting graphs to paths and trees. Algorithmica, 68(1):109-132, 2014. URL:
  28. Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov, and Christophe Paul. Obtaining a bipartite graph by contracting few edges. SIAM J. Discret. Math., 27(4):2143-2156, 2013. URL:
  29. Falk Hüffner, Christian Komusiewicz, Hannes Moser, and Rolf Niedermeier. Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst., 47(1):196-217, 2010. Google Scholar
  30. Takehiro Ito, Marcin Kaminski, Daniël Paulusma, and Dimitrios M. Thilikos. Parameterizing cut sets in a graph by the number of their components. Theor. Comput. Sci., 412(45):6340-6350, 2011. URL:
  31. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL:
  32. R. Krithika, Pranabendu Misra, and Prafullkumar Tale. An FPT algorithm for contraction to cactus. In Proceedings of the 24th International Conference on Computing and Combinatorics, pp. 341-352, 2018. Google Scholar
  33. R. Krithika, Roohani Sharma, and Prafullkumar Tale. The complexity of contracting bipartite graphs into small cycles. CoRR, abs/2206.07358, 2022. URL:
  34. Bingkai Lin. The parameterized complexity of the k-biclique problem. J. ACM, 65(5):34:1-34:23, 2018. URL:
  35. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. On the hardness of eliminating small induced subgraphs by contracting edges. In Proceedings of the 8th International Symposium on Parameterized and Exact Computation, pp. 243-254, 2013. Google Scholar
  36. Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 224-237. ACM, 2017. URL:
  37. Saket Saurabh, Uéverton dos Santos Souza, and Prafullkumar Tale. On the parameterized complexity of grid contraction. In 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT, volume 162 of LIPIcs, pages 34:1-34:17, 2020. URL:
  38. Thomas J. Schaefer. The complexity of satisfiability problems. In Richard J. Lipton, Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho, editors, Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pages 216-226. ACM, 1978. URL:
  39. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discret. Appl. Math., 3(2):151-153, 1981. Google Scholar
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