We call a depth-$4$ formula $C$ $\textit{ set-depth-4}$ if there exists a (unknown) partition $X_1\sqcup\cdots\sqcup X_d$ of the variable indices $[n]$ that the top product layer respects, i.e. $C(\mathbf{x})=\sum_{i=1}^k {\prod_{j=1}^{d} {f_{i,j}(\mathbf{x}_{X_j})}}$ $ ,$ where $f_{i,j}$ is a $\textit{sparse}$ polynomial in $\mathbb{F}[\mathbf{x}_{X_j}]$. Extending this definition to any depth - we call ... more >>>

Given an instance of a hard decision problem, a limited goal is to $compress$ that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas $\psi^1, \ldots, \psi^t$, we must determine if at least one $\psi^j$ is satisfiable. An $OR-compression ... more >>>

Convex relaxations based on different hierarchies of
linear/semi-definite programs have been used recently to devise
approximation algorithms for various optimization problems. The
approximation guarantee of these algorithms improves with the number
of {\em rounds} $r$ in the hierarchy, though the complexity of solving
(or even writing down the solution for) ... more >>>