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 >>>

Border complexity captures polynomials that can be approximated arbitrarily well by small algebraic circuits, and debordering asks how efficiently such an approximation can be converted into an exact computation. Debordering lies at the heart of the gap between Valiant's determinant versus permanent conjecture and its strengthening by Mulmuley and Sohoni ... more >>>