Sparse Sets, Approximable Sets, and Parallel Queries to NP

We show that if an NP-complete set or a coNP-complete set is polynomial-time disjunc-tive truth-table 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 NP-complete set or a coNP-complete set is disjunctively reducible to a sparse set of polylogarithmic density then P = NP.