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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.

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