# 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}).

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