Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels

Authors Akanksha Agrawal, M. S. Ramanujan

Thumbnail PDF


  • Filesize: 0.79 MB
  • 17 pages

Document Identifiers

Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, India
M. S. Ramanujan
  • University of Warwick, Coventry, UK

Cite AsGet BibTex

Akanksha Agrawal and M. S. Ramanujan. Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels. 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. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The problem of computing a minimum set of vertices intersecting a finite set of forbidden minors in a given graph is a fundamental graph problem in the area of kernelization with numerous well-studied special cases. A major breakthrough in this line of research was made by Fomin et al. [FOCS 2012], who showed that the ρ-Treewidth Modulator problem (delete minimum number of vertices to ensure that treewidth is at most ρ) has a polynomial kernel of size k^g(ρ) for some function g. A second standout result in this line is that of Giannapoulou et al. [ACM TALG 2017], who obtained an f(η)k^𝒪(1)-size kernel (for some function f) for the η-Treedepth Modulator problem (delete fewest number of vertices to make treedepth at most η) and showed that some dependence of the exponent of k on ρ in the result of Fomin et al. for the ρ-Treewidth Modulator problem is unavoidable under reasonable complexity hypotheses. In this work, we provide an approximate interpolation between these two results by giving, for every ε > 0, a (1+ε)-approximate kernel of size f'(η,ρ,1/ε)⋅ k^g'(ρ) (for some functions f' and g') for the problem of deciding whether k vertices can be deleted from a given graph to obtain a graph that has elimination distance at most η to the class of graphs that have treewidth at most ρ. Graphs of treedepth η are precisely the graphs with elimination distance at most η-1 to the graphs of treewidth 0 and graphs of treewidth ρ are simply graphs with elimination distance 0 to graphs of treewidth ρ. Consequently, our result "approximately" interpolates between these two major results in this active line of research.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Lossy Kernelization
  • Treewidth Modulator
  • Vertex Deletion Problems


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


  1. Akanksha Agrawal and M. S. Ramanujan. On the parameterized complexity of clique elimination distance. In Yixin Cao and Marcin Pilipczuk, editors, 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 180 of LIPIcs, pages 1:1-1:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  2. Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi. Grundy distinguishes treewidth from pathwidth. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 14:1-14:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  3. Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica, 75(2):363-382, 2016. Google Scholar
  4. Jannis Bulian and Anuj Dawar. Fixed-parameter tractable distances to sparse graph classes. Algorithmica, 79(1):139-158, 2017. URL:
  5. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL:
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  7. Reinhard Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 3rd edition, 2005. Google Scholar
  8. Rodney G Downey and Michael Ralph Fellows. Parameterized complexity. Springer Science & Business Media, 2012. Google Scholar
  9. Samuel Fiorini, Gwenaël Joret, and Ugo Pietropaoli. Hitting diamonds and growing cacti. In Integer Programming and Combinatorial Optimization, 14th International Conference, IPCO 2010, Lausanne, Switzerland, June 9-11, 2010. Proceedings, pages 191-204, 2010. URL:
  10. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM J. Discrete Math., 30(1):383-410, 2016. URL:
  11. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 470-479, 2012. URL:
  12. Archontia C. Giannopoulou, P. Jansen Bart M. Daniel Lokshtanov, and Saket Saurabh. Uniform kernelization complexity of hitting forbidden minors. ACM Trans. Algorithms, 13(3):35:1-35:35, 2017. URL:
  13. Bart M. P. Jansen, Jari J. H. de Kroon, and Michal Wlodarczyk. Vertex deletion parameterized by elimination distance and even less. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1757-1769. ACM, 2021. URL:
  14. Gwenaël Joret, Christophe Paul, Ignasi Sau, Saket Saurabh, and Stéphan Thomassé. Hitting and harvesting pumpkins. SIAM J. Discrete Math., 28(3):1363-1390, 2014. URL:
  15. Tuukka Korhonen. Single-exponential time 2-approximation algorithm for treewidth. In FOCS, 2021. Google Scholar
  16. Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. Google Scholar
  17. Alexander Lindermayr, Sebastian Siebertz, and Alexandre Vigny. Elimination distance to bounded degree on planar graphs. In Javier Esparza and Daniel Král', editors, 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 65:1-65:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  18. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization - Preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond, pages 129-161. Springer, 2012. Google Scholar
  19. Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. CoRR, abs/1604.04111, 2016. URL:
  20. Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. Lossy kernelization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 224-237, 2017. URL:
  21. Dániel Marx and Ildikó Schlotter. Obtaining a planar graph by vertex deletion. Algorithmica, 62(3-4):807-822, 2012. URL:
  22. Geevarghese Philip, Venkatesh Raman, and Yngve Villanger. A quartic kernel for pathwidth-one vertex deletion. In Graph Theoretic Concepts in Computer Science - 36th International Workshop, WG 2010, Zarós, Crete, Greece, June 28-30, 2010 Revised Papers, pages 196-207, 2010. URL: