Skip to main content
SpringerLink
Log in

Computing Hitting Set Kernels By AC0-Circuits

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

Given a hypergraph H = (V,E), what is the smallest subset \(X \subseteq V\) such that e ∩ X≠∅ holds for all e ∈ E? This problem, known as the hitting set problem, is a basic problem in combinatorial optimization and has been studied extensively in both classical and parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph H and a number k as input and output a hypergraph H′ such that (1) H has a hitting set of size k if and only if \(H^{\prime }\) has such a hitting set and (2) the size of \(H^{\prime }\) depends only on k and on the maximum cardinality d of hyperedges in H. The algorithms run in polynomial time and can be parallelized to a certain degree: one can easily compute hitting set kernels in parallel time O(k) and not-so-easily in time O(d) – but it was conjectured that these are the best parallel algorithms possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Abu-Khzam, F.N., Langston, M.A., Shanbhag, P., Symons, C.T.: Scalable parallel algorithms for fpt problems. Algorithmica 45(3), 269–284 (2006)

    Article  MathSciNet  Google Scholar 

  2. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)

    Article  MathSciNet  Google Scholar 

  3. Bannach, M., Stockhusen, C., Tantau, T.: Fast parallel fixed-parameter algorithms via color coding. In: Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC 2015), pp. 224–235 (2015)

  4. Bannach, M., Tantau, T.: Parallel multivariate meta-theorems. In: Proceedings of the 11th International Symposium on Parameterized and Exact Computation (IPEC 2016), pp. 4:1–4:17 (2016)

  5. Bannach, M., Tantau, T.: Computing Kernels in parallel: lower and upper bounds. In: Proceedings of the 13th International Symposium on Parameterized and Exact Computation (IPEC 2018), pp. 13:1–13:14 (2018)

  6. Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC1. In: Proceedings of the 3rd Annual Structure in Complexity Theory Conference, pp. 47–59 (1988)

  7. Cai, L., Chen, J., Downey, R. G., Fellows, M.R.: Advice classes of parameterized tractability. Ann. Pure Appl. Logic 84(1), 119–138 (1997)

    Article  MathSciNet  Google Scholar 

  8. Cesati, M., Di Ianni, M.: Parameterized parallel complexity. In: Proceedings of the 4th International Euro-Par Conference, vol. 1470 of Lecture Notes in Computer Science, pp. 892–896. Springer (1998)

  9. Chen, Y., Flum, J.: Some lower bounds in parameterized AC0. In: Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), pp. 27:1–27:14 (2016)

  10. Chen, Y., Flum, J., Grohe, M.: Bounded nondeterminism and alternation in parameterized complexity theory. In: Proceedings of the 18th IEEE Conference on Computational Complexity (CCC 2003), pp. 13–29. IEEE Computer Society (2003)

  11. Chen, Y., Flum, J., Huang, X.: Slicewise definability in first-order logic with bounded quantifier rank. In: Proceedings of the 26th EACSL Annual Conference on Computer Science Logic, (CSL 2017), August 20–24. Stockholm, Sweden, pp. 19:1–19:16 (2017)

  12. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput. 24, 873–921 (1995)

    Article  MathSciNet  Google Scholar 

  13. Elberfeld, M., Stockhusen, C., Tantau, T.: On the space complexity of parameterized problems: Classes and completness. Algorithmica 71(3), 661–701 (2014)

    Article  Google Scholar 

  14. Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 1(1), 85–90 (1960)

    Article  MathSciNet  Google Scholar 

  15. Flum, J., Grohe, M.: Describing parameterized complexity classes. In: Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2002), vol. 2285 of Lecture Notes in Computer Science, pp. 359–371. Springer (2002)

  16. Flum, J., Grohe, M.: Parameterized complexity theory. Texts in Theoretical Computer Science. Springer (2006)

  17. Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algoritms 1(1), 89–102 (2003)

    Article  MathSciNet  Google Scholar 

  18. Pilipczuk, M., Siebertz, S., Toruńczyk, S.: Parameterized circuit complexity of model-checking on sparse structures. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, (LICS 2018), pp 789–798. ACM, New York (2018)

  19. van Bevern, R.: Towards optimal and expressive kernelization for d-hitting set. Algorithmica 70(1), 129–147 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Computer Science, Universität zu Lübeck, Lübeck, Germany

    Max Bannach & Till Tantau

Authors
  1. Max Bannach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Till Tantau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Till Tantau.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the Topical Collection on Special Issue on Theoretical Aspects of Computer Science (2018)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bannach, M., Tantau, T. Computing Hitting Set Kernels By AC0-Circuits. Theory Comput Syst 64, 374–399 (2020). https://doi.org/10.1007/s00224-019-09941-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-019-09941-z

Keywords

Search

Navigation