Abstract
We reduce the problem of detecting the existence of an object to the problem of computing the parity of the number of objects in question. In particular, when given any non-empty set system, we prove that randomly restricting elements of its ground set makes the size of the restricted set system an odd number with significant probability. When compared to previously known reductions of this type, ours excel in their simplicity: For graph problems, restricting elements of the ground set usually corresponds to simple deletion and contraction operations, which can be encoded efficiently in most problems. We find three applications of our reductions:
-
1.
An exponential-time algorithm: We show how to decide Hamiltonicity in directed \(n\)-vertex graphs with running time \(1.9999^n\) provided that the graph has at most \(1.0385^n\) Hamiltonian cycles. We do so by reducing to the algorithm of Björklund and Husfeldt (FOCS 2013) that computes the parity of the number of Hamiltonian cycles in time \(1.619^n\).
-
2.
A new result in the framework of Cygan et al. (CCC 2012) for analyzing the complexity of NP-hard problems under the Strong Exponential Time Hypothesis: If the parity of the number of Set Covers can be determined in time \(1.9999^n\), then Set Cover can be decided in the same time.
-
3.
A structural result in parameterized complexity: We define the parameterized complexity class \(\oplus \)W[1] and prove that it is at least as hard as W[1] under randomized fpt-reductions with bounded one-sided error; this is analogous to the classical result \(\mathrm {NP\subseteq RP^{\oplus P}}\) by Toda (SICOMP 1991).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)
Björklund, A., Husfeldt, T.: The parity of directed hamiltonian cycles. In: Proc. 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, Berkeley, CA, USA, October 26–29, pp. 727–735 (2013)
Björklund, A.: Below all subsets for permutational counting problems, (2012) arXiv:1211.0391 [cs:DS]
Björklund, A.: Determinant sums for undirected Hamiltonicity. SIAM J. Comput. 43(1), 280–299 (2014)
Björklund, A., Husfeldt, T., Lyckberg, I.: Computing the permanent modulo a prime power. In: preparation (2015)
Calabro, C., Impagliazzo, R., Kabanets, V., Paturi, R.: The complexity of unique k-SAT: An isolation lemma for k-CNFs. In: Proc. 18th IEEE Conference on Computational Complexity, CCC, Aarhus, Denmark, July 7–10 (2003)
Cohen, G., Tal, A.: Two structural results for low degree polynomials and applications. Electronic Colloquium on Computational Complexity (ECCC). Tech report TR13-145 (2013)
Cygan, M., Kratsch, S., Nederlof, J.: Fast Hamiltonicity checking via bases of perfect matchings. In: Proc. 45th Symposium on Theory of Computing, STOC, Palo Alto, CA, USA, June 1–4, pp. 301–310 (2013)
Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlström, M.: On problems as hard as CNFSAT. In: Proc. 27th IEEE Conference on Computational Complexity, CCC, Porto, Portugal, June 26–84 (2012)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)
Gupta, S.: Isolating an odd number of elements and applications in complexity theory. Theor. Comput. Syst. 31(1), 27–40 (1998)
Montoya, J.A., Müller, M.: Parameterized random complexity. Theor. Comput. Syst. 52(2), 221–270 (2013)
Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987)
Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)
Traxler, P.: The time complexity of constraint satisfaction. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 190–201. Springer, Heidelberg (2008)
Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 85–93 (1986)
Williams, V.V., Wang, J., Williams, R., Yu, H.: Finding four-node subgraphs in triangle time. In: Proc. 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, San Diego, CA, USA, January 4–6, 2015, pp. 1671–1680 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Björklund, A., Dell, H., Husfeldt, T. (2015). The Parity of Set Systems Under Random Restrictions with Applications to Exponential Time Problems. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_19
Download citation
DOI: https://doi.org/10.1007/978-3-662-47672-7_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-47671-0
Online ISBN: 978-3-662-47672-7
eBook Packages: Computer ScienceComputer Science (R0)