Which Problems Have Strongly Exponential Complexity?

For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that CircuitSAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, Vertex Cover, are SERF–complete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC0; that is, bounds of the form 2 (n). This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even limited (at most n ) fan-in for bottom-level gates would imply a nonlinear size lower bound for logarithmic depth circuits. We show that with high probability even degree 2 random GF(2) polynomials require strongly exponential size for k3 circuits for k = o(log logn). We thus exhibit a much smaller space of 2O(n2) functions such that almost every function in this class requires strongly exponential size k3 circuits. As a corollary, we derive a pseudorandom generator (requiring O(n2) bits of advice) that maps n bits into a larger number of bits so that computing parity on the range is hard for k3 circuits. Our main technical lemma is an algorithm that, for any fixed > 0, represents an arbitrary k-CNF formula as a disjunction of 2 n k-CNF formulas that are sparse, that is, each has O(n) clauses.