The Parity of Set Systems Under Random Restrictions with Applications to Exponential Time Problems

  • Andreas Björklund
  • Holger DellEmail author
  • Thore Husfeldt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


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. 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. 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. 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).



Success Probability Hamiltonian Cycle Main Lemma Random Restriction Extremal Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Björklund, A.: Below all subsets for permutational counting problems, (2012) arXiv:1211.0391 [cs:DS]
  4. 4.
    Björklund, A.: Determinant sums for undirected Hamiltonicity. SIAM J. Comput. 43(1), 280–299 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Björklund, A., Husfeldt, T., Lyckberg, I.: Computing the permanent modulo a prime power. In: preparation (2015)Google Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Cohen, G., Tal, A.: Two structural results for low degree polynomials and applications. Electronic Colloquium on Computational Complexity (ECCC). Tech report TR13-145 (2013)Google Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  11. 11.
    Gupta, S.: Isolating an odd number of elements and applications in complexity theory. Theor. Comput. Syst. 31(1), 27–40 (1998)zbMATHCrossRefGoogle Scholar
  12. 12.
    Montoya, J.A., Müller, M.: Parameterized random complexity. Theor. Comput. Syst. 52(2), 221–270 (2013)zbMATHCrossRefGoogle Scholar
  13. 13.
    Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    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) CrossRefGoogle Scholar
  16. 16.
    Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 85–93 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Andreas Björklund
    • 1
  • Holger Dell
    • 2
    Email author
  • Thore Husfeldt
    • 1
    • 3
  1. 1.Lund UniversityLundSweden
  2. 2.Saarland University and Cluster of Excellence (MMCI)SaarbruckenGermany
  3. 3.IT University of CopenhagenCopenhagenDenmark

Personalised recommendations