Parameterized Valiant’s Classes

Authors Markus Bläser, Christian Engels

Thumbnail PDF


  • Filesize: 478 kB
  • 14 pages

Document Identifiers

Author Details

Markus Bläser
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Christian Engels
  • IIT Bombay, Mumbai, India


We thank Holger Dell for valuable comments on a first draft of this paper and Radu Curticapean and Marc Roth for helpful discussions.

Cite AsGet BibTex

Markus Bläser and Christian Engels. Parameterized Valiant’s Classes. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to the permanent with vastly different parameterized complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Algebraic complexity theory
  • parameterized complexity theory
  • Valiant’s classes


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


  1. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci., 87:119-139, 2017. URL:
  2. Peter Bürgisser. Completeness and Reduction in Algebraic Complexity Theory. Springer, 2000. Google Scholar
  3. Peter Bürgisser. Cook’s versus Valiant’s hypothesis. Theor. Comput. Sci., 235(1):71-88, 2000. URL:
  4. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Applied Mathematics, 108(1-2):23-52, 2001. URL:
  5. Radu Curticapean. Counting Matchings of Size k Is W[1]-Hard. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 352-363. Springer, 2013. URL:
  6. Radu Curticapean, Holger Dell, Fedor V. Fomin, Leslie Ann Goldberg, and John Lapinskas. A Fixed-Parameter Perspective on #BIS. CoRR, abs/1702.05543, 2017. URL:
  7. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. 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 210-223. ACM, 2017. URL:
  8. Radu Curticapean and Dániel Marx. Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 130-139. IEEE Computer Society, 2014. URL:
  9. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  10. Uffe Flarup, Pascal Koiran, and Laurent Lyaudet. On the Expressive Power of Planar Perfect Matching and Permanents of Bounded Treewidth Matrices. In Takeshi Tokuyama, editor, Algorithms and Computation, 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17-19, 2007, Proceedings, volume 4835 of Lecture Notes in Computer Science, pages 124-136. Springer, 2007. URL:
  11. Jörg Flum and Martin Grohe. The Parameterized Complexity of Counting Problems. SIAM J. Comput., 33(4):892-922, 2004. URL:
  12. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL:
  13. Anna Galluccio and Martin Loebl. On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents. Electr. J. Comb., 6, 1999. URL:
  14. Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. Parameterized Complexity of Vertex Cover Variants. Theory Comput. Syst., 41(3):501-520, 2007. URL:
  15. Christian Ikenmeyer and Stefan Mengel. On the relative power of reduction notions in arithmetic circuit complexity. Inf. Process. Lett., 130:7-10, 2018. URL:
  16. Mark Jerrum and Kitty Meeks. The parameterised complexity of counting even and odd induced subgraphs. Combinatorica, 37(5):965-990, 2017. URL:
  17. Pascal Koiran and Sylvain Perifel. VPSPACE and a transfer theorem over the complex field. In Ludek Kucera and Antonín Kucera, editors, Mathematical Foundations of Computer Science 2007, 32nd International Symposium, MFCS 2007, Ceský Krumlov, Czech Republic, August 26-31, 2007, Proceedings, volume 4708 of Lecture Notes in Computer Science, pages 359-370. Springer, 2007. URL:
  18. Pascal Koiran and Sylvain Perifel. VPSPACE and a transfer theorem over the reals. In Wolfgang Thomas and Pascal Weil, editors, STACS 2007, 24th Annual Symposium on Theoretical Aspects of Computer Science, Aachen, Germany, February 22-24, 2007, Proceedings, volume 4393 of Lecture Notes in Computer Science, pages 417-428. Springer, 2007. URL:
  19. Meena Mahajan. Algebraic Complexity Classes. CoRR, abs/1307.3863, 2013. URL:
  20. Meena Mahajan and B. V. Raghavendra Rao. Small-Space Analogues of Valiant’s Classes. In Miroslaw Kutylowski, Witold Charatonik, and Maciej Gebala, editors, Fundamentals of Computation Theory, 17th International Symposium, FCT 2009, Wroclaw, Poland, September 2-4, 2009. Proceedings, volume 5699 of Lecture Notes in Computer Science, pages 250-261. Springer, 2009. URL:
  21. Meena Mahajan and Nitin Saurabh. Some Complete and Intermediate Polynomials in Algebraic Complexity Theory. In Alexander S. Kulikov and Gerhard J. Woeginger, editors, Computer Science - Theory and Applications - 11th International Computer Science Symposium in Russia, CSR 2016, St. Petersburg, Russia, June 9-13, 2016, Proceedings, volume 9691 of Lecture Notes in Computer Science, pages 251-265. Springer, 2016. URL:
  22. Gerhard Ringel. Das Geschlecht des vollständigen paaren Graphen. Abh. Math. Sem. Univ. Hamburg 28, pages 139-150, 1965. Google Scholar
  23. Marc Roth. Counting Restricted Homomorphisms via Möbius Inversion over Matroid Lattices. In Kirk Pruhs and Christian Sohler, editors, 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, volume 87 of LIPIcs, pages 63:1-63:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL:
  24. Leslie G. Valiant. Completeness Classes in Algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL:
  25. Virginia Vassilevska and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 455-464. ACM, 2009. 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