, Volume 75, Issue 2, pp 403–423 | Cite as

AND-Compression of NP-Complete Problems: Streamlined Proof and Minor Observations

  • Holger DellEmail author


Drucker (Proceedings of the 53rd annual symposium on foundations of computer science (FOCS), pp 609–618. doi: 10.1109/FOCS.2012.712012) proved the following result: Unless the unlikely complexity-theoretic collapse \(\mathsf {coNP}\subseteq \mathsf {NP/poly}\) occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a streamlined proof of Drucker’s theorem. An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances \(x_1,\ldots ,x_t\) to a single SAT-instance y of size \({\text {poly}}(\max _i|x_i|)\) such that y is satisfiable if and only if all \(x_i\) are satisfiable. The “AND” in the name stems from the fact that the predicate “y is satisfiable” can be written as the AND of all predicates “\(x_i\) is satisfiable”. Drucker’s theorem complements the result by Bodlaender et al. (J Comput Syst Sci 75:423–434, 2009) and Fortnow and Santhanam (J Comput Syst Sci 77:91–106, 2011), who proved the analogous statement for OR-compressions, and Drucker’s proof not only subsumes their result but also extends it to randomized compression algorithms that are allowed to have a certain probability of failure. Drucker (Proceedings of the 53rd annual symposium on foundations of computer science (FOCS), pp 609–618. doi: 10.1109/FOCS.2012.712012) presented two proofs: The first uses information theory and the minimax theorem from game theory, and the second is an elementary, iterative proof that is not as general. In our proof, we realize the iterative structure as a generalization of the arguments of Ko (J Comput Syst Sci 26:209–211, 1983) for \(\mathsf {P}\)-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. Our proof achieves the full generality of Drucker’s theorem, avoids the minimax theorem, and restricts the use of information theory to a single, intuitive lemma about the average noise sensitivity of compressive maps. To prove this lemma, we use the same information-theoretic inequalities as Drucker.


Boolean Function Polynomial Kernel Statistical Distance Leibler Divergence Boolean Circuit 
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.



I would like to thank Andrew Drucker, Martin Grohe, and Noy Galil Rotbart for encouraging me to pursue the publication of this manuscript, David Xiao for pointing out Theorem 2 to me, Andrew Drucker, Dániel Marx, and anonymous referees for comments on an earlier version of this paper, and Dieter van Melkebeek for some helpful discussions.


  1. 1.
    Drucker, A.: New limits to classical and quantum instance compression. In: Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 609–618 (2012). doi: 10.1109/FOCS.2012.71
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75, 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77, 91–106 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ko, K.-I.: On self-reducibility and weak P-selectivity. J. Comput. Syst. Sci. 26, 209–211 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Harnik, D., Naor, M.: On the compressibility of NP instances and cryptographic applications. SIAM J. Comput. 39, 1667–1713 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61 (2014). doi: 10.1145/2629620
  7. 7.
    Dell, H., Kabanets, V., van Melkebeek, D., Watanabe, O.: Is Valiant-Vazirani’s isolation probability improvable? Comput. Complex. 22, 345–383 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Xiao, D.: New perspectives on the complexity of computational learning, and other problems in theoretical computer science Ph.D. thesis. Princeton University. (2009)
  9. 9.
    Drucker, A.: New limits to classical and quantum instance compression tech report TR12-112 rev. 3 (Electronic Colloquium on Computational Complexity (ECCC). (2014)
  10. 10.
    Dell, H., Marx, D.: Kernelization of packing problems. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 68–81 (2012). doi: 10.1137/1.9781611973099.6
  11. 11.
    Hermelin, D., Wu, X.: Weak compositions and their applications to polynomial lower bounds for kernelization. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 104–113 (2012). doi: 10.1137/1.9781611973099.9
  12. 12.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28, 277–305 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kratsch, S.: Co-nondeterminism in compositions: a kernelization lower bound for a Ramsey-type problem. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 114–122 (2012). doi: 10.1137/1.9781611973099.10
  14. 14.
    Kratsch, S., Philip, G., Ray, S.: Point line cover: the easy kernel is essentially tight. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1596–1606 (2014). doi: 10.1137/1.9781611973402.116
  15. 15.
    Arora, S., Barak, B.: Computational Complexity—A Modern Approach, Cambridge University Press. ISBN: 978-0-521-42426-4 (2009)Google Scholar
  16. 16.
    Adleman, L.M.: Two theorems on random polynomial time. In: Proceedings of the 19th Annual Symposium on Foundations of Computer Science (FOCS), pp. 75–83 (1978). doi: 10.1109/SFCS.1978.37
  17. 17.
    Sahai, A., Vadhan, S.: A complete problem for statistical zero knowledge. J. ACM 50, 196–249 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Goldreich, O., Vadhan, S.: On the complexity of computational problemsregarding distributions (a survey) Tech report TR11-004 ElectronicColloquium on Computational Complexity (ECCC). (2011)
  19. 19.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, London (2012)zbMATHGoogle Scholar
  20. 20.
    Fedotov, A.A., Harremoës, P., Topsoe, F.: Refinements of Pinsker’s inequality. IEEE Trans. Inf. Theory 49, 1491–1498 (2003)CrossRefzbMATHGoogle Scholar
  21. 21.
    Reid, M., Williamson, B.: Generalised Pinsker inequalities. In: Proceedings of the 22nd Annual Conference on Learning Theory (COLT), pp. 18–21. (2009)

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Cluster of Excellence (MMCI)Saarland UniversitySaarbrückenGermany
  2. 2.Simons Institute for the Theory of ComputingBerkeleyUSA
  3. 3.UC BerkeleyBerkeleyUSA
  4. 4.LIAFAUniversité Paris DiderotParisFrance

Personalised recommendations