Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
An Improved Exponential-Time Algorithm for k-SAT
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
An algorithm for the satisfiability problem of formulas in conjunctive normal form
Journal of Algorithms
Theoretical Computer Science
A Duality between Clause Width and Clause Density for SAT
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Boolean satisfiability from theoretical hardness to practical success
Communications of the ACM - A Blind Person's Interaction with Technology
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Improving exhaustive search implies superpolynomial lower bounds
Proceedings of the forty-second ACM symposium on Theory of computing
Fighting Perebor: New and Improved Algorithms for Formula and QBF Satisfiability
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Non-uniform ACC Circuit Lower Bounds
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
A satisfiability algorithm for AC0
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for k-CNFs: every k-CNF is a sub-exponential size disjunction of k-CNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. A natural question is whether an analogous structural result holds for CNFs or even for broader non-uniform classes such as constant-depth circuits or Boolean formulae. We prove a very strong negative result in this connection: For every superlinear function f(n), there are CNFs of size f(n) which cannot be written as a disjunction of 2n−εn CNFs each having a linear number of clauses for any ε0. We also give a hierarchy of such non-sparsifiable CNFs: For every k, there is a k′ for which there are CNFs of size nk′ which cannot be written as a sub-exponential size disjunction of CNFs of size nk. Furthermore, our lower bounds hold not just against CNFs but against an arbitrary family of functions as long as the cardinality of the family is appropriately bounded. As by-products of our result, we make progress both on questions about circuit lower bounds for depth-3 circuits and satisfiability algorithms for constant-depth circuits. Improving on a result of Impagliazzo, Paturi and Zane, for any f(n)=ω(n log(n)), we define a pseudo-random function generator with seed length f(n) such that with high probability, a function in the output of this generator does not have depth-3 circuits of size 2n−o(n) with bounded bottom fan-in. We show that if we could decrease the seed length of our generator below n, we would get an explicit function which does not have linear-size logarithmic-depth series-parallel circuits, solving a long-standing open question. Motivated by the question of whether CNFs sparsify into bounded-depth circuits, we show a simplification result for bounded-depth circuits: any bounded-depth circuit of linear size can be written as a sub-exponential size disjunction of linear-size constant-width CNFs. As a corollary, we show that if there is an algorithm for CNF satisfiability which runs in time O(2αn) for some fixed αO(2(α+o(1))n).