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On the existence of hard sparse sets under weak reductions

  • Jin -Yi Cai
  • Ashish V. Naik
  • D. Sivakumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)

Abstract

Recently a 1978 conjecture by Hartmanis was resolved by Cai and Sivakumar, following progress made by Ogihara. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P=LOGSPACE. We extend these results to the case of sparse sets that are hard under more general reducibilities. Furthermore, the proof technique can be applied to resolve open questions about hard sparse sets for NP as well. Using algebraic and probabilistic techniques, we show the following results.

  1. (1)

    If there exists a sparse set that is hard for P under bounded truthtable reductions, then P=NC2.

     
  2. (2)

    If there exists a sparse set that is hard for P under randomized logspace reductions with two-sided error, then P=RLOGSPACE (with two-way access to the random tape).

     
  3. (3)

    If there exists an NP-hard sparse set under randomized polynomial-time reductions with two-sided error, then NP=RP.

     
  4. (4)

    If there exists a disjunctive truth-table hard sparse set for NP, then NP=RP.

     

Keywords

Boolean Circuit Random Tape Table Reduction SIGACT News Fast Parallel Algorithm 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Jin -Yi Cai
    • 1
  • Ashish V. Naik
    • 2
  • D. Sivakumar
    • 3
  1. 1.Department of Computer ScienceState University of New York at BuffaloBuffalo
  2. 2.Computer Science DepartmentUniversity of ChicagoChicago
  3. 3.Department of Computer ScienceState University of New York at BuffaloBuffalo

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