Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion

Authors Bart M. P. Jansen , Ruben F. A. Verhaegh

Thumbnail PDF


  • Filesize: 0.9 MB
  • 17 pages

Document Identifiers

Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Ruben F. A. Verhaegh
  • Eindhoven University of Technology, The Netherlands

Cite AsGet BibTex

Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


For an optimization problem Π on graphs whose solutions are vertex sets, a vertex v is called c-essential for Π if all solutions of size at most c ⋅ opt contain v. Recent work showed that polynomial-time algorithms to detect c-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size k of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected n-vertex graph G in time 2^𝒪(𝓁³)⋅n^𝒪(1), where 𝓁 is the number of vertices in an optimal solution that are not 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of c, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that (2-ε)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Linear programming
  • Theory of computation → Rounding techniques
  • Theory of computation → Fixed parameter tractability
  • fixed-parameter tractability
  • essential vertices
  • integrality gap


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


  1. Tobias Achterberg, Robert E. Bixby, Zonghao Gu, Edward Rothberg, and Dieter Weninger. Presolve reductions in mixed integer programming. Technical Report 16-44, ZIB, Takustr.7, 14195 Berlin, 2016. URL:
  2. Manuel Aprile, Matthew Drescher, Samuel Fiorini, and Tony Huynh. A tight approximation algorithm for the cluster vertex deletion problem. Math. Program., 197(2):1069-1091, 2023. URL:
  3. Nicolas Bousquet, Jean Daligault, and Stéphan Thomassé. Multicut is FPT. SIAM J. Comput., 47(1):166-207, 2018. URL:
  4. Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-space reduction via essential vertices. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, Proceedings of the 30th Annual European Symposium on Algorithms, ESA 2022, volume 244 of LIPIcs, pages 30:1-30:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. 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. Parameterized complexity of vertex colouring. Discret. Appl. Math., 127(3):415-429, 2003. URL:
  7. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Comput. Complex., 15(2):94-114, 2006. URL:
  8. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM, 55(5):21:1-21:19, 2008. URL:
  9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  10. Rodney G. Downey and M. R. Fellows. Parameterized Complexity. Springer Publishing Company, Incorporated, 2012. Google Scholar
  11. Matthew Drescher. Two Approaches to Approximation Algorithms for Vertex Deletion Problems. PhD thesis, Université libre de bruxelles, 2021. URL:
  12. Michael R. Fellows. The lost continent of polynomial time: Preprocessing and kernelization. In Proceedings of the 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006, pages 276-277, 2006. URL:
  13. Henning Fernau. A top-down approach to search-trees: Improved algorithmics for 3-hitting set. Algorithmica, 57(1):97-118, 2010. URL:
  14. Fedor Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  15. Daniel Golovin, Viswanath Nagarajan, and Mohit Singh. Approximating the k-multicut problem. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, pages 621-630. ACM Press, 2006. URL:
  16. Jens Gramm, Jiong Guo, Falk Hüffner, and Rolf Niedermeier. Automated generation of search tree algorithms for hard graph modification problems. Algorithmica, 39(4):321-347, 2004. URL:
  17. Meike Hatzel, Lars Jaffke, Paloma T. Lima, Tomás Masarík, Marcin Pilipczuk, Roohani Sharma, and Manuel Sorge. Fixed-parameter tractability of DIRECTED MULTICUT with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 3229-3244. SIAM, 2023. URL:
  18. Pinar Heggernes, Pim van 't Hof, Bart M. P. Jansen, Stefan Kratsch, and Yngve Villanger. Parameterized complexity of vertex deletion into perfect graph classes. Theor. Comput. Sci., 511:172-180, 2013. URL:
  19. Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Comb., 21(1):39-60, 2001. URL:
  20. Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-space reduction via essential vertices revisited: Vertex multicut and cograph deletion, 2024. URL:
  21. Subhash Khot. On the power of unique 2-prover 1-round games. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, STOC 2002, pages 767-775. ACM, 2002. URL:
  22. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci., 74(3):335-349, 2008. URL:
  23. Monique Laurent. A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0-1 programming. Math. Oper. Res., 28(3):470-496, 2003. URL:
  24. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci., 20(2):219-230, 1980. URL:
  25. Daniel Lokshtanov, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. FPT-approximation for FPT problems. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 199-218. SIAM, 2021. URL:
  26. Dániel Marx and Igor Razgon. Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput., 43(2):355-388, 2014. URL:
  27. James Nastos and Yong Gao. Bounded search tree algorithms for parametrized cograph deletion: Efficient branching rules by exploiting structures of special graph classes. Discret. Math. Algorithms Appl., 4(1), 2012. URL:
  28. Rolf Niedermeier and Peter Rossmanith. An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algorithms, 1(1):89-102, 2003. URL:
  29. Marcin Pilipczuk and Magnus Wahlström. Directed multicut is W[1]-hard, even for four terminal pairs. ACM Trans. Comput. Theory, 10(3):13:1-13:18, 2018. URL:
  30. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1-33:38, 2019. URL:
  31. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  32. Ola Svensson. Hardness of vertex deletion and project scheduling. Theory Comput., 9:759-781, 2013. URL:
  33. Vijay V. Vazirani. Approximation algorithms. Springer, 2001. URL:
  34. Karsten Weihe. Covering trains by stations or the power of data reduction. In Algorithms and Experiments (ALEX98), pages 1-8, 1998. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail