On the Limits of Sparsification

Abstract

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 2 n − εn CNFs each having a linear number of clauses for any ε > 0. We also give a hierarchy of such non-sparsifiable CNFs: For every k, there is a k′ for which there are CNFs of size n k′ which cannot be written as a sub-exponential size disjunction of CNFs of size n k . 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 2 n − 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(2 αn ) for some fixed α < 1 on CNFs of linear size, then there is an algorithm for satisfiability of linear-size constant-depth circuits which runs in time O(2(α + o(1))n ).