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Sparse Sets, Approximable Sets, and Parallel Queries to NP
 Proc. Sixteenth Symposium on Theoretical Aspects of Computing (STACS '99), LNCS 1563
, 1999
"... We show that if an NPcomplete set or a coNPcomplete set is polynomialtime disjunctive truthtable reducible to a sparse set then FP NP jj = FP NP [log]. With a similar argument we show also that if SAT is O(log n)approximable then FP NP jj = FP NP [log]. Since FP NP jj = FP NP [lo ..."
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We show that if an NPcomplete set or a coNPcomplete set is polynomialtime disjunctive truthtable reducible to a sparse set then FP NP jj = FP NP [log]. With a similar argument we show also that if SAT is O(log n)approximable then FP NP jj = FP NP [log]. Since FP NP jj = FP NP [log] implies that SAT is O(logn)approximable [BFT97], it follows from our result that the two hypotheses are equivalent. We also show that if an NPcomplete set or a coNPcomplete set is disjunctively reducible to a sparse set of polylogarithmic density then P = NP. 1 Introduction The study of the existence of sparse hard sets for complexity classes has occupied complexity theorists for over two decades. The first results in this area were motivated by the BermanHartmanis isomorphism conjecture [BH77] and by the study of connections between uniform and nonuniform complexity classes [KL80]. The focus shifted to proving, for various reducibilities 1 (whose strengths lie between the many...