On the Parameterized Complexity of Maximum Degree Contraction Problem

Authors Saket Saurabh, Prafullkumar Tale

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Saket Saurabh
  • The Institute Of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Prafullkumar Tale
  • CISPA - Helmholtz Center for Information Security, Saarbrücken, Germany


We want to thank the anonymous reviewers for their valuable feedback.

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Saket Saurabh and Prafullkumar Tale. On the Parameterized Complexity of Maximum Degree Contraction Problem. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time n^𝒪(k). As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an FPT algorithm running in time (d + k)^(2k) ⋅ n^𝒪(1) = 2^𝒪(k log (k+d)) ⋅ n^𝒪(1), and remains NP-hard for every constant d ≥ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^𝒪(dk) ⋅ n^𝒪(1). In particular, our algorithm runs in time 2^𝒪(k) ⋅ n^𝒪(1), for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by k + d. We answer this question in the negative and prove that it does not admit a polynomial compression unless NP ⊆ coNP/poly.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Graph Contraction Problems
  • FPT Algorithm
  • Lower Bound
  • ETH
  • No Polynomial Kernel


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  1. Akanksha Agarwal, Saket Saurabh, and Prafullkumar Tale. On the parameterized complexity of contraction to generalization of trees. Theory of Computing Systems, 63(3):587-614, 2019. Google Scholar
  2. Akanksha Agrawal, Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. Path contraction faster than 2ⁿ. SIAM Journal on Discrete Mathematics, 34(2):1302-1325, 2020. Google Scholar
  3. Akanksha Agrawal, Lawqueen Kanesh, Saket Saurabh, and Prafullkumar Tale. Paths to trees and cacti. In International Conference on Algorithms and Complexity, pages 31-42. Springer, 2017. Google Scholar
  4. Akanksha Agrawal, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Split contraction: The untold story. ACM Transactions on Computation Theory (TOCT), 11(3):1-22, 2019. Google Scholar
  5. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM (JACM), 42(4):844-856, 1995. Google Scholar
  6. Takao Asano and Tomio Hirata. Edge-contraction problems. Journal of Computer and System Sciences, 26(2):197-208, 1983. Google Scholar
  7. Balabhaskar Balasundaram, Shyam Sundar Chandramouli, and Svyatoslav Trukhanov. Approximation algorithms for finding and partitioning unit-disk graphs into co-k-plexes. Optimization Letters, 4(3):311-320, 2010. Google Scholar
  8. Rémy Belmonte, Petr A. Golovach, Pim Hof, and Daniël Paulusma. Parameterized complexity of three edge contraction problems with degree constraints. Acta Informatica, 51(7):473-497, 2014. Google Scholar
  9. Nadja Betzler, Robert Bredereck, Rolf Niedermeier, and Johannes Uhlmann. On bounded-degree vertex deletion parameterized by treewidth. Discrete Applied Mathematics, 160(1-2):53-60, 2012. Google Scholar
  10. Hans L Bodlaender and Babette van Antwerpen-de Fluiter. Reduction algorithms for graphs of small treewidth. Information and Computation, 167(2):86-119, 2001. Google Scholar
  11. Andries Evert Brouwer and Henk Jan Veldman. Contractibility and NP-completeness. Journal of Graph Theory, 11(1):71-79, 1987. Google Scholar
  12. Leizhen Cai and Chengwei Guo. Contracting few edges to remove forbidden induced subgraphs. In International Symposium on Parameterized and Exact Computation, pages 97-109. Springer, 2013. Google Scholar
  13. Zhi-Zhong Chen, Michael Fellows, Bin Fu, Haitao Jiang, Yang Liu, Lusheng Wang, and Binhai Zhu. A linear kernel for co-path/cycle packing. In International Conference on Algorithmic Applications in Management, pages 90-102. Springer, 2010. Google Scholar
  14. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  15. Anders Dessmark, Klaus Jansen, and Andrzej Lingas. The maximum k-dependent and f-dependent set problem. In International Symposium on Algorithms and Computation, pages 88-97. Springer, 1993. Google Scholar
  16. Michael R Fellows, Jiong Guo, Hannes Moser, and Rolf Niedermeier. A generalization of nemhauser and trotter’s local optimization theorem. Journal of Computer and System Sciences, 77(6):1141-1158, 2011. Google Scholar
  17. Fedor V. Fomin, Daniel Lokshtanov, Ivan Mihajlin, Saket Saurabh, and Meirav Zehavi. Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), pages 49:1-49:18. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  18. Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On structural parameterizations of the bounded-degree vertex deletion problem. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  19. Petr A Golovach, Marcin Kaminski, Daniël Paulusma, and Dimitrios M Thilikos. Increasing the minimum degree of a graph by contractions. Theoretical computer science., 481:74-84, 2013. Google Scholar
  20. Petr A Golovach, Pim van’t Hof, and Daniël Paulusma. Obtaining planarity by contracting few edges. Theoretical Computer Science, 476:38-46, 2013. Google Scholar
  21. Sylvain Guillemot and Dániel Marx. A faster fpt algorithm for bipartite contraction. Information Processing Letters, 113(22-24):906-912, 2013. Google Scholar
  22. Spoorthy Gunda, Pallavi Jain, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. On the parameterized approximability of contraction to classes of chordal graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020), 2020 (To Appear). Google Scholar
  23. Pinar Heggernes, Pim Van'T Hof, Daniel Lokshtanov, and Christophe Paul. Obtaining a bipartite graph by contracting few edges. SIAM Journal on Discrete Mathematics, 27(4):2143-2156, 2013. Google Scholar
  24. 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. Google Scholar
  25. Christian Komusiewicz, Falk Hüffner, Hannes Moser, and Rolf Niedermeier. Isolation concepts for efficiently enumerating dense subgraphs. Theoretical Computer Science, 410(38-40):3640-3654, 2009. Google Scholar
  26. R Krithika, Pranabendu Misra, and Prafullkumar Tale. An fpt algorithm for contraction to cactus. In International Computing and Combinatorics Conference, pages 341-352. Springer, 2018. Google Scholar
  27. Ramaswamy Krithika, Pranabendu Misra, Ashutosh Rai, and Prafullkumar Tale. Lossy kernels for graph contraction problems. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  28. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. SIAM Journal on Computing, 47(3):675-702, 2018. Google Scholar
  29. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. On the hardness of eliminating small induced subgraphs by contracting edges. In International Symposium on Parameterized and Exact Computation, pages 243-254. Springer, 2013. Google Scholar
  30. Barnaby Martin and Daniël Paulusma. The computational complexity of disconnected cut and 2k2-partition. Journal of combinatorial theory, series B, 111:17-37, 2015. Google Scholar
  31. Naomi Nishimura, Prabhakar Ragde, and Dimitrios M Thilikos. Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover. Discrete Applied Mathematics, 152(1-3):229-245, 2005. Google Scholar
  32. 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 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  33. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discrete Applied Mathematics, 3(2):151-153, 1981. Google Scholar
  34. Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the np-hardness of edge-deletion and-contraction problems. Discrete Applied Mathematics, 6(1):63-78, 1983. Google Scholar