Counting Problems in Parameterized Complexity

Author Radu Curticapean

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Radu Curticapean
  • Basic Algorithms Research Copenhagen (BARC) and IT University of Copenhagen, Copenhagen, Denmark

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Radu Curticapean. Counting Problems in Parameterized Complexity. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs. While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way.

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
  • counting complexity
  • parameterized complexity
  • graph motifs
  • perfect matchings
  • graph minor theory
  • Hamiltonian cycles


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  1. Noga Alon, Phuong Dao, Iman Hajirasouliha, Fereydoun Hormozdiari, and S Cenk Sahinalp. Biomolecular network motif counting and discovery by color coding. Bioinformatics, 24(13):i241-i249, 2008. URL:
  2. Noga Alon and Shai Gutner. Balanced Hashing, Color Coding and Approximate Counting. In Parameterized and Exact Computation, 4th International Workshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected Papers, pages 1-16, 2009. URL:
  3. Noga Alon and Shai Gutner. Balanced families of perfect hash functions and their applications. ACM Trans. Algorithms, 6(3):54:1-54:12, 2010. URL:
  4. Noga Alon, Raphael Yuster, and Uri Zwick. Color-Coding. J. ACM, 42(4):844-856, 1995. URL:
  5. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy Problems for Tree-Decomposable Graphs. J. Algorithms, 12(2):308-340, 1991. URL:
  6. Vikraman Arvind and Venkatesh Raman. Approximation Algorithms for Some Parameterized Counting Problems. In Algorithms and Computation, 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings, pages 453-464, 2002. URL:
  7. Andreas Björklund. Determinant Sums for Undirected Hamiltonicity. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 173-182, 2010. URL:
  8. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set Partitioning via Inclusion-Exclusion. SIAM Journal on Computing, 39(2):546-563, 2009. URL:
  9. Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Counting Thin Subgraphs via Packings Faster than Meet-in-the-Middle Time. ACM Trans. Algorithms, 13(4):48:1-48:26, 2017. URL:
  10. Markus Bläser and Radu Curticapean. Weighted Counting of k-Matchings Is #W[1]-Hard. In Parameterized and Exact Computation - 7th International Symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012. Proceedings, pages 171-181, 2012. URL:
  11. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL:
  12. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (Meta) Kernelization. J. ACM, 63(5):44:1-44:69, 2016. URL:
  13. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 151-164, 2018. URL:
  14. Cornelius Brand, Holger Dell, and Marc Roth. Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP. In 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, pages 9:1-9:14, 2016. URL:
  15. Cornelius Brand and Marc Roth. Parameterized Counting of Trees, Forests and Matroid Bases. In Computer Science - Theory and Applications - 12th International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8-12, 2017, Proceedings, pages 85-98, 2017. URL:
  16. Andrei Bulatov and Martin Grohe. The complexity of partition functions. Theoretical Computer Science, 348(2):148-186, 2005. URL:
  17. Andrei A. Bulatov. The complexity of the counting constraint satisfaction problem. J. ACM, 60(5):34, 2013. URL:
  18. Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330, 2017. URL:
  19. Jin-Yi Cai and Xi Chen. Complexity of Counting CSP with Complex Weights. J. ACM, 64(3):19:1-19:39, 2017. URL:
  20. Jin-yi Cai and Pinyan Lu. Holographic algorithms: From art to science. J. Comput. Syst. Sci., 77(1):41-61, 2011. URL:
  21. Jin-yi Cai, Pinyan Lu, and Mingji Xia. Computational Complexity of Holant Problems. SIAM J. Comput., 40(4):1101-1132, 2011. URL:
  22. Hubie Chen and Stefan Mengel. Counting Answers to Existential Positive Queries: A Complexity Classification. In Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2016, San Francisco, CA, USA, June 26 - July 01, 2016, pages 315-326, 2016. URL:
  23. Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W. Juedes, Iyad A. Kanj, and Ge Xia. Tight lower bounds for certain parameterized NP-hard problems. Information and Computation, 201(2):216-231, 2005. URL:
  24. Xi Chen, Martin E. Dyer, Leslie Ann Goldberg, Mark Jerrum, Pinyan Lu, Colin McQuillan, and David Richerby. The complexity of approximating conservative counting CSPs. J. Comput. Syst. Sci., 81(1):311-329, 2015. URL:
  25. Yijia Chen, Marc Thurley, and Mark Weyer. Understanding the Complexity of Induced Subgraph Isomorphisms. In Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, pages 587-596, 2008. URL:
  26. 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:
  27. Radu Curticapean. Counting Matchings of Size k Is #W[1]-Hard. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 352-363, 2013. URL:
  28. Radu Curticapean. Counting perfect matchings in graphs that exclude a single-crossing minor. CoRR, abs/1406.4056, 2014. URL:,
  29. Radu Curticapean. Block interpolation: A framework for tight exponential-time counting complexity. Inf. Comput., 261(Part):265-280, 2018. URL:
  30. Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for counting small subgraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 210-223, 2017. URL:
  31. Radu Curticapean, Holger Dell, and Marc Roth. Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes. In 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, pages 25:1-25:15, 2017. URL:
  32. Radu Curticapean, Nathan Lindzey, and Jesper Nederlof. A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1080-1099, 2018. URL:
  33. 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, 2014. URL:
  34. Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1650-1669, 2016. URL:
  35. Radu Curticapean and Mingji Xia. Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 994-1009, 2015. URL:
  36. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  37. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast Hamiltonicity Checking Via Bases of Perfect Matchings. J. ACM, 65(3):12:1-12:46, 2018. URL:
  38. Paul Dagum and Michael Luby. Approximating the Permanent of Graphs with Large Factors. Theoretical Computer Science, 102(2):283-305, 1992. URL:
  39. Víctor Dalmau and Peter Jonsson. The complexity of counting homomorphisms seen from the other side. Theoretical Computer Science, 329(1-3):315-323, 2004. URL:
  40. Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlen. Exponential Time Complexity of the Permanent and the Tutte Polynomial. ACM Transactions on Algorithms, 10(4):21, 2014. URL:
  41. E. Demaine, M. Hajiaghayi, and D. Thilikos. Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors. Algorithmica, 41(4):245-267, 2005. URL:
  42. Erik D. Demaine and MohammadTaghi Hajiaghayi. The Bidimensionality Theory and Its Algorithmic Applications. Comput. J., 51(3):292-302, 2008. URL:
  43. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  44. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL:
  45. Martin Dyer, Leslie Ann Goldberg, and Mike Paterson. On counting homomorphisms to directed acyclic graphs. J. ACM, 54, December 2007. URL:
  46. Martin E. Dyer and Catherine S. Greenhill. The complexity of counting graph homomorphisms. Random Struct. Algorithms, 17(3-4):260-289, 2000. URL:;2-W.
  47. Martin E. Dyer and David Richerby. An Effective Dichotomy for the Counting Constraint Satisfaction Problem. SIAM J. Comput., 42(3):1245-1274, 2013. URL:
  48. Jörg Flum and Martin Grohe. The parameterized complexity of counting problems. SIAM Journal on Computing, 33(4):892-922, 2004. URL:
  49. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  50. Fedor V. Fomin, Erik D. Demaine, and Mohammad Taghi Hajiaghayi. Bidimensionality. In Encyclopedia of Algorithms. Springer, 2015. URL:
  51. Fedor V. Fomin and Dieter Kratsch. Subset Convolution, pages 125-139. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. URL:
  52. Markus Frick. Generalized Model-Checking over Locally Tree-Decomposable Classes. Theory Comput. Syst., 37(1):157-191, 2004. URL:
  53. Anna Galluccio and Martin Loebl. On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents. Electronic Journal of Combinatorics, 6, 1998. Google Scholar
  54. Leslie Ann Goldberg and Mark Jerrum. Approximating the Tutte polynomial of a binary matroid and other related combinatorial polynomials. J. Comput. Syst. Sci., 79(1):68-78, 2013. URL:
  55. Leslie Ann Goldberg and Mark Jerrum. The Complexity of Approximately Counting Tree Homomorphisms. TOCT, 6(2):8, 2014. URL:
  56. Mark S. Granovetter. The Strength of Weak Ties. American Journal of Sociology, 78(6):1360-1380, 1973. Google Scholar
  57. Thore Husfeldt. Invitation to Algorithmic Uses of Inclusion-Exclusion. In Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part II, pages 42-59, 2011. URL:
  58. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. System Sci., 63(4):512-530, 2001. Google Scholar
  59. A. Isihara. Statistical physics. Academic Press, 1971. Google Scholar
  60. M. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM, 51(4):671-697, 2004. URL:
  61. Mark Jerrum and Kitty Meeks. Some hard families of parameterized counting problems. ACM Transactions on Computation Theory, 7(3):11, 2015. URL:
  62. Mark Jerrum and Kitty Meeks. The parameterised complexity of counting connected subgraphs and graph motifs. J. Comput. Syst. Sci., 81(4):702-716, 2015. URL:
  63. Mark Jerrum and Kitty Meeks. The parameterised complexity of counting even and odd induced subgraphs. Combinatorica, pages 1-26, 2016. URL:
  64. Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on Computing, 22(5):1087-1116, 1993. URL:
  65. Pieter W. Kasteleyn. The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica, 27(12):1209-1225, 1961. URL:
  66. Pieter W. Kasteleyn. Graph Theory and Crystal Physics. In Graph Theory and Theoretical Physics, pages 43-110. Academic Press, 1967. Google Scholar
  67. Gustav Kirchhoff. Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik und Chemie, LXXIL(12), 1847. Google Scholar
  68. Tomer Kotek and Johann A. Makowsky. Connection Matrices and the Definability of Graph Parameters. Logical Methods in Computer Science, 10(4), 2014. URL:
  69. Miroslaw Kowaluk, Andrzej Lingas, and Eva-Marta Lundell. Counting and Detecting Small Subgraphs via Equations. SIAM J. Discrete Math., 27(2):892-909, 2013. URL:
  70. Charles Little. An Extension of Kasteleyn’s method of enumerating the 1-factors of planar graphs. In Combinatorial Mathematics, LNCS, pages 63-72. Springer, 1974. URL:
  71. László Lovász. Operations with structures. Acta Mathematica Hungarica, 18(3-4):321-328, 1967. Google Scholar
  72. László Lovász. The rank of connection matrices and the dimension of graph algebras. Eur. J. Comb., 27(6):962-970, 2006. URL:
  73. László Lovász. Large networks and graph limits, volume 60. American Mathematical Society Providence, 2012. Google Scholar
  74. Johann A. Makowsky. Algorithmic uses of the Feferman-Vaught Theorem. Annals of Pure and Applied Logic, 126(1-3):159-213, 2004. Provinces of logic determined. Essays in the memory of Alfred Tarski. Parts I, II and III. URL:
  75. Johann A. Makowsky, Udi Rotics, Ilya Averbouch, and Benny Godlin. Computing Graph Polynomials on Graphs of Bounded Clique-Width. In Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers, pages 191-204, 2006. URL:
  76. Dániel Marx. Can You Beat Treewidth? Theory of Computing, 6(1):85-112, 2010. URL:
  77. Catherine McCartin. Parameterized counting problems. Ann. Pure Appl. Logic, 138(1-3):147-182, 2006. URL:
  78. Kitty Meeks. The challenges of unbounded treewidth in parameterised subgraph counting problems. Discrete Applied Mathematics, 198:170-194, 2016. URL:
  79. Ron Milo, Shai Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri Chklovskii, and Uri Alon. Network motifs: Simple building blocks of complex networks. Science, 298(5594):824-827, 2002. URL:
  80. Bojan Mohar. A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface. SIAM J. Discrete Math., 12(1):6-26, 1999. URL:
  81. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 26(2):415-419, 1985. Google Scholar
  82. Rolf Niedermeier. Invitation to fixed-parameter algorithms, volume 31 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2006. Google Scholar
  83. Neil Robertson and Paul D. Seymour. Excluding a graph with one crossing. In Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991, pages 669-675, 1993. Google Scholar
  84. Neil Robertson and Paul D. Seymour. Graph Minors. XVI. Excluding a non-planar graph. J. Comb. Theory, Ser. B, 89(1):43-76, 2003. URL:
  85. Marc Roth. Counting Restricted Homomorphisms via Möbius Inversion over Matroid Lattices. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 63:1-63:14, 2017. URL:
  86. Marc Roth and Johannes Schmitt. Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness. CoRR, abs/1807.01920, 2018. URL:
  87. Alexander D. Scott and Gregory B. Sorkin. Linear-programming design and analysis of fast algorithms for Max 2-CSP. Discrete Optimization, 4(3-4):260-287, 2007. URL:
  88. Richard P. Stanley. Enumerative Combinatorics, volume 1. Cambridge University Press, second edition, 2011. Google Scholar
  89. Simon Straub, Thomas Thierauf, and Fabian Wagner. Counting the Number of Perfect Matchings in K₅-Free Graphs. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 66-77, 2014. URL:
  90. H. N. V. Temperley and Michael E. Fisher. Dimer problem in statistical mechanics - an exact result. Philosophical Magazine, 6(68):1478-6435, 1961. Google Scholar
  91. Marc Thurley. Kernelizations for Parameterized Counting Problems. In Theory and Applications of Models of Computation, 4th International Conference, TAMC 2007, Shanghai, China, May 22-25, 2007, Proceedings, pages 703-714, 2007. URL:
  92. Salil P. Vadhan. The Complexity of Counting in Sparse, Regular, and Planar Graphs. SIAM J. Comput., 31(2):398-427, 2001. URL:
  93. Leslie G. Valiant. The complexity of computing the permanent. Theoret. Comput. Sci., 8(2):189-201, 1979. Google Scholar
  94. Leslie G. Valiant. Holographic Algorithms. SIAM J. Comput., 37(5):1565-1594, 2008. URL:
  95. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution. In Algorithms - ESA 2009, 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings, pages 566-577, 2009. URL:
  96. Vijay V. Vazirani. NC algorithms for computing the number of perfect matchings in K_3,3-free graphs and related problems. Inf. Comput., 80(2):152-164, 1989. Google Scholar
  97. Virginia Vassilevska Williams and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. SIAM Journal on Computing, 42(3):831-854, 2013. URL:
  98. Mingji Xia, Peng Zhang, and Wenbo Zhao. Computational complexity of counting problems on 3-regular planar graphs. Theoretical Computer Science, 384(1):111-125, 2007. Theory and Applications of Models of Computation. URL:
  99. Dmitriy Zhuk. A Proof of CSP Dichotomy Conjecture. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342, 2017. URL: