Lov\'{a}sz and Schrijver introduced several lift and project methods for $0$-$1$ integer programs, now collectively known as Lov\'{a}sz-Schrijver ($LS$) hierarchies. Several lower bounds have since been proven for the rank of various linear programming relaxations in the $LS$ and $LS_+$ hierarchies. In this paper we investigate rank bounds in the ... more >>>

We propose an abstract framework for studying search-to-decision reductions for NP. Specifically, we study the following witness finding problem: for a hidden nonempty set $W\subseteq\{0,1\}^n$, the goal is to output a witness in $W$ with constant probability by making randomized queries of the form ``is $Q\cap W$ nonempty?''\ where $Q\subseteq\{0,1\}^n$. ... more >>>

Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, ... more >>>