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Which problems have strongly exponential complexity? (English) Zbl 1006.68052
Summary: 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 algorithm for these problems. We introduce a generalized reduction that we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that Circuit-SAT is SERF-complete for all NP-search problems, and that for any fixed $$k\geq 3$$, $$k$$-SAT, $$k$$-Colorability, $$k$$-Set Cover, Independent Set, Clique, and 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 AC$$^0$$, that is, bounds of the form $$2^{\Omega(n)}$$. This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even limited (at most $$n^\varepsilon$$) fan-in for bottom-level gates would imply a nonlinear size lower bound for logarithmic depth circuits. We show that with high probability even random degree 2 GF(2) polynomials require strongly exponential size for $$\Sigma^k_3$$ circuits for $$k= o(\log\log n)$$. We thus exhibit a much smaller space of $$2^{O(n^2)}$$ functions such that almost every function in this class requires strongly exponential size $$\Sigma^k_3$$ circuits. As a corollary, we derive a pseudorandom generator (requiring $$O(n^2)$$ bits of advice) that maps $$n$$ bits into a larger number of bits so that computing parity on the range is hard for $$\Sigma^k_3$$ circuits. Our main technical lemma is an algorithm that, for any fixed $$\varepsilon> 0$$, represents an arbitrary $$k$$-CNF formula as a disjunction of $$2^{\varepsilon n}$$ $$k$$-CNF formulas that are sparse, that is, each disjunct has $$O(n)$$ clauses.

MSC:
 68Q25 Analysis of algorithms and problem complexity 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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