Abstract
We define instance compressibility ([5,7]) for parametric problems in PH and PSPACE. We observe that the problem Σ i CircuitSAT of deciding satisfiability of a quantified Boolean circuit with i − 1 alternations of quantifiers starting with an existential quantifier is complete for parametric problems in the class \(\Sigma_{i}^{p}\) with respect to W-reductions, and that analogously the problem QBCSAT (Quantified Boolean Circuit Satisfiability) is complete for parametric problems in PSPACE with respect to W-reductions. We show the following results about these problems:
-
1
If CircuitSAT is non-uniformly compressible within NP, then Σ i CircuitSAT is non-uniformly compressible within NP, for any i ≥ 1.
-
2
If QBCSAT is non-uniformly compressible (or even if satisfiability of quantified Boolean CNF formulae is non-uniformly compressible), then PSPACE ⊆ NP/poly and PH collapses to the third level.
Next, we define Succinct Interactive Proof (Succinct IP) and by adapting the proof of IP = PSPACE ([4,2]), we show that QBFormulaSAT (Quantified Boolean Formula Satisfiability) is in Succinct IP. On the contrary if QBFormulaSAT has Succinct PCPs ([11]), Polynomial Hierarchy (PH) collapses.
Keywords
- Polynomial Time
- Parametric Problem
- Boolean Formula
- Input String
- Compression Function
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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the ACM 28(1), 114–133 (1981)
Shamir, A.: IP = PSPACE. Journal of the ACM 39(4), 869–877 (1992)
Yap, C.K.: Some consequences of non-uniform conditions on uniform classes. Theoretical Computer Science 26, 287–300 (1983)
Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. Journal of the ACM 39(4), 859–868 (1992)
Harnik, D., Naor, M.: On the compressibility of NP instances and cryptographic applications. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 719–728 (2006)
Buhrman, H., Hitchcock, J.M.: NP-Hard Sets are Exponentially Dense Unless NP is contained in coNP/poly. Elect. Colloq. Comput. Complex (ECCC) 15(022) (2008)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)
Abrahamson, K.A., Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness IV: On completeness for W[P] and PSPACE analogs. Annals of pure and applied logic 73, 235–276 (1995)
Babai, L., Moran, S.: Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences 36, 254–276 (1988)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. Journal of Computer and System Sciences 77(1), 91–106 (2011); special issues celebrating Karp’s Kyoto Prize
Sipser, M.: Introduction to the Theory of Computation. Course Technology, 2nd edn. (2005)
Karp, R.M., Lipton, R.J.: Some connections between nonuniform and uniform complexity classes. In: Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, pp. 302–309 (1980), doi:10.1145/800141.804678
Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press (2006)
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)
Goldwasser, S., Micali, S., Rackoff, C.: The Knowledge complexity of interactive proof-systems. In: Proceedings of 17th ACM Symposium on the Theory of Computation, Providence, Rhode Island, pp. 291–304 (1985)
Kratsch, S., Wahlström, M.: Preprocessing of Min Ones Problems: A Dichotomy. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 653–665. Springer, Heidelberg (2010)
Chen, Y., Flum, J., Muller, M.: Lower bounds for kernelizations. CRM Publications (November 2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chakraborty, C., Santhanam, R. (2012). Instance Compression for the Polynomial Hierarchy and beyond. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-33293-7_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33292-0
Online ISBN: 978-3-642-33293-7
eBook Packages: Computer ScienceComputer Science (R0)
