Sparse sets, approximable sets, and parallel queries to NP

Authors:
Vikraman Arvind;Jacobo Torán
Affiliations:
Institute of Mathematical Sciences, Chennai, India;Abteilung Theoretische Informatik, Universität Ulm, Ulm, Germany
Venue:
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
Year:
1999

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Abstract

We show that if an NP-complete set or a coNP-complete set is polynomial-time disjunctive truth-table reducible to a sparse set then FP||NP = FPNP[log]. Similarly, we show that if SAT is O(log n)- approximable then FP||NP = FPNP[log]. Since FP||NP = FPNP[log] implies that SAT is O(log n)-approximable [BFT97], it follows from our result that these two hypotheses are equivalent. We also show that if an NP-complete set or a coNP-complete set is disjunctively reducible to a sparse set of polylogarithmic density then, in fact, P = NP.