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Optimization Problems in the Polynomial-Time Hierarchy

  • Christopher Umans
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)

Abstract

This talk surveys work on classifying the complexity and approximability of problems residing in the Polynomial-Time Hierarchy, above the first level. Along the way, we highlight some prominent natural problems that are believed – but not yet known – to be \(\Sigma^p_2\)-complete. We describe how strong inapproximability results for certain \(\Sigma^p_2\) optimization problems can be obtained using dispersers to build error-correcting codes. Finally we adapt a learning algorithm to produce approximation algorithms for these problems.

Keywords

Approximation Algorithm Boolean Circuit SIGACT News Constant Depth Circuit Circuit Lower Bound 
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.

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References

  1. Stockmeyer, L.J.: The polynomial-time hierarchy. Theoretical Computer Science 3(1), 1–22 (1976)CrossRefMathSciNetGoogle Scholar
  2. Schaefer, M., Umans, C.: Completeness in the polynomial-time hierarchy: a compendium. In: SIGACT News (guest Complexity Theory column) (2002)Google Scholar
  3. Schaefer, M., Umans, C.: Completeness in the polynomial-time hierarchy: Part II. In: SIGACT News (guest Complexity Theory column) (2002)Google Scholar
  4. Umans, C.: The Minimum Equivalent DNF Problem and Shortest Implicants. Journal of Computer and System Sciences 63(4), 597–611 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. Umans, C.: Hardness of Approximating Σp 2 Minimization Problems. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 465–474 (1999)Google Scholar
  6. Mossel, E., Umans, C.: On the complexity of approximating the VC dimension. Journal of Computer and System Sciences 65(4), 660–671 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. Shaltiel, R.: Recent developments in explicit constructions of extractors. BEATCS Computational Complexity Column 77 (2002)Google Scholar
  8. Srinivasan, A., Zuckerman, D.: Computing with very weak random sources. SIAM Journal on Computing 28(4), 1433–1459 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. Ta-Shma, A., Umans, C., Zuckerman, D.: Loss-less condensers, unbalanced expanders, and extractors. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 143–152. ACM Press, New York (2001)Google Scholar
  10. Bellare, M., Goldreich, O., Petrank, E.: Uniform generation of NP-witnesses using an NP-oracle. Inf. Comput. 163(2), 510–526 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. Bshouty, N.H., Cleve, R., Gavaldà, R., Kannan, S., Tamon, C.: Oracles and queries that are sufficient for exact learning. Journal of Computer and System Sciences 52(3), 421–433 (1996)MATHCrossRefMathSciNetGoogle Scholar
  12. Umans, C.: Approximability and Completeness in the Polynomial Hierarchy. PhD thesis, U.C. Berkeley (2000)Google Scholar
  13. Hemaspaandra, E., Wechsung, G.: The Minimization Problem for Boolean Formulas. In: 38th Annual Symposium on Foundations of Computer Science, pp. 575–584. IEEE, Florida (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christopher Umans
    • 1
  1. 1.Computer Science DepartmentCalifornia Institute of TechnologyPasadenaUSA

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