A constant-depth circuit consists of AND and OR gates of unbounded fan-in, arranged in a constant number of layers, that read the input literals; for example, a depth-three circuit is an OR of CNFs or an AND of DNFs. The classical lower bound of Håstad (1989) states that every depth-$d$ ... more >>>

We give a simplified proof of the polynomial improvements for integer 3SUM and all-pairs shortest paths (APSP) established by Alman and Vassilevska Williams, with weaker exponents. The central ingredient is selected-entry matrix multiplication: computing a prescribed set of entries of a product of dense rectangular matrices. Starting from Sch\"onhage's inner/outer-product ... more >>>

A pruning procedure maps a Boolean circuit to a circuit on the same variables that accepts only satisfying assignments of the original.
Valiant and Vazirani give a pruning procedure that leaves exactly one satisfying assignment of every satisfiable circuit with probability $\Omega(1/n)$, where $n$ is the number of variables. Dell, ... more >>>