Abstract
In this chapter, we will first look at parameterized counting problems as an analog to the classical problem of counting. We establish the classes #W[t] and related issues, and prove completeness results. We present the Flum–Grohe results on the hardness of counting k-cycles. Later we introduce a formal model for parameterized randomization. We prove an analog of the Valiant–Vazirani Theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
That is, the terms of each row form a geometric progression. These are well studied and known to always be nonsingular, with a standard expression for the determinant.
- 2.
Here we are choosing f uniformly at random.
References
A. Andrzejak, An algorithm for the Tutte polynomials of graphs of bounded treewidth. Discrete Math. 190(1–3), 39–54 (1998)
V. Arvind, V. Raman, Approximation algorithms for some parameterized counting problems, in Algorithms and Computation: Proceedings of 13th International Symposium, ISAAC 2002, Vancouver, BC, Canada, November 21–23, 2002, ed. by P. Bose, P. Morin. LNCS, vol. 2518 (Springer, Berlin, 2002), pp. 453–464
L. Cai, S.M. Chan, S.O. Chan, Random separation: a new method for solving fixed-cardinality optimization problems, in Parameterized and Exact Computation. Proceedings of Second International Workshop, IWPEC ’06, Zürich, Switzerland, September 13–15, 2006, ed. by H. Bodlaender, M. Langston. LNCS, vol. 4169 (Springer, Berlin, 2006), pp. 239–250
L. Cai, B. Yang, Parameterized complexity of even/odd subgraph problems, in 7th International Conference on Algorithms and Complexity, CIAC 2010, Rome, Italy, May 26–28, 2010, ed. by T. Calamoneri, J. Diaz. LNCS, vol. 6078 (Springer, Berlin, 2010), pp. 85–96
R. Chitnis, M. Cygan, M. Hajiaghayi, M. Pilipczuk, M. Pilipczuk, Designing FPT algorithms for cut problems using randomized contractions, in IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, New Jersey, USA, October 20–23, 2012 (IEEE Comput. Soc., Los Alamitos, 2012), pp. 460–469
B. Courcelle, J. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)
M. Cygan, Deterministic parameterized connected vertex cover, arXiv:1202.6642
P. Damaschke, Degree-bounded techniques accelerate some parameterized graph algorithms, in Parameterized and Exact Computation, Proceedings of 4th International Workshop, IWPEC ’09, Copenhagen, Denmark, September 2009, ed. by J. Chen, F. Fomin. LNCS, vol. 5917 (Springer, Berlin, 2009), pp. 98–109
R. Downey, M. Fellows, K. Regan, Parameterized circuit complexity and the W hierarchy. Theor. Comput. Sci. 191(1–2), 97–115 (1998)
J. Flum, M. Grohe, The parameterized complexity of counting problems, in Proceedings of 43rd Symposium on Foundations of Computer Science, FOCS 2002, Vancouver, BC, Canada, 16–19 November 2002 (IEEE Comput. Soc., Los Alamitos, 2002), pp. 538–547
M. Frick, M. Grohe, Deciding first-order properties of locally tree-decomposable graphs, in Proceedings of 26th International Colloquium on Automata, Languages and Programming (ICALP 1999), Prague, Czech Republic, July 11–15, 1999, ed. by J. Wiedermann, P. van Emde Boas, M. Nielsen. LNCS, vol. 1644 (Springer, Berlin, 1999), pp. 331–340
B. Lin, Y. Chen, The parameterized complexity of k-edge induced subgraphs, in Proceedings of 39th International Colloquium on Automata, Languages and Programming (ICALP 2012), Part I, Warick, UK, July 9–13, 2012, ed. by A. Czumaj, K. Mehlhorn, A. Pitts, R. Wattenhofer. LNCS, vol. 7391 (Springer, Berlin, 2012), pp. 641–652
D. Marx, D. Lokstanov, Clustering with local restrictions. Inf. Comput. 222, 278–292 (2013)
C. McCartin, Contributions to parameterized complexity, PhD thesis, Victoria University, Wellington, 2002
C. McCartin, Parameterized counting problems. Ann. Pure Appl. Log. 138(1–3), 147–182 (2006)
A. Montoya, The parameterized complexity of probability amplification. Inf. Process. Lett. 109(1), 46–53 (2008)
J.A. Montoya, M. Müller, Parameterized random complexity. Theory Comput. Syst. 52(2), 221–270 (2013)
M. Müller, Valiant–Vazirani lemmata for various logics, Technical Report 63, Electronic Colloquium on Computational Complexity, 2008
S. Noble, Evaluation of the Tutte polynomial for graphs of bounded tree-width. Comb. Probab. Comput. 7, 307–321 (1998)
M. Thurley, Kernelizations for parameterized counting problems, in Theory and Applications of Models of Computation, Proceedings of 4th International Conference, TAMC 2007, Shanghai, China, May 22–25, 2007, ed. by J. Cai, B. Cooper, H. Zhu. LNCS, vol. 4484 (Springer, Berlin, 2007), pp. 703–714
L. Valiant, The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979)
D. Welsh, A. Gale, The complexity of counting problems, in Aspects of Complexity, ed. by R. Downey, D. Hirschfeldt. De Gruyter Series in Logic and Its Applications (de Gruyter, Berlin, 2001), pp. 115–154
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
Downey, R.G., Fellows, M.R. (2013). Parameterized Counting and Randomization. In: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-5559-1_32
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5559-1_32
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5558-4
Online ISBN: 978-1-4471-5559-1
eBook Packages: Computer ScienceComputer Science (R0)