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On the existence of hard sparse sets under weak reductions

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STACS 96 (STACS 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1046))

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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.

Research supported in part by NSF grants CCR-9057486 and CCR-9319093, and an Alfred P. Sloan Fellowship.

Research supported in part by NSF grant CCR 92-53582.

Research supported in part by NSF grant CCR-9409104.

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Author information

Authors and Affiliations

  1. Department of Computer Science, State University of New York at Buffalo, 14260, Buffalo, NY

    Jin -Yi Cai

  2. Computer Science Department, University of Chicago, 60637, Chicago, IL

    Ashish V. Naik

  3. Department of Computer Science, State University of New York at Buffalo, 14260, Buffalo, NY

    D. Sivakumar

Authors
  1. Jin -Yi Cai
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  2. Ashish V. Naik
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  3. D. Sivakumar
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Editor information

Claude Puech Rüdiger Reischuk

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© 1996 Springer-Verlag Berlin Heidelberg

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Cai, J.Y., Naik, A.V., Sivakumar, D. (1996). On the existence of hard sparse sets under weak reductions. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_26

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  • DOI: https://doi.org/10.1007/3-540-60922-9_26

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60922-3

  • Online ISBN: 978-3-540-49723-3

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