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We give a pseudorandom generator that fools $m$-facet polytopes over ${0,1}^n$ with seed length $mathrm{polylog}(m) cdot log n$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any ${0,1}$-integer program.
Say that A is a Hadamard factorization of the identity I_n of size n if the entrywise product of A and the transpose of A is I_n. It can be easily seen that the rank of any Hadamard factorization of the identity must be at least sqrt{n}. Dietzfelbing
We give a pseudorandom generator that fools degree-$d$ polynomial threshold functions over $n$-dimensional Gaussian space with seed length $mathrm{poly}(d)cdot log n$. All previous generators had a seed length with at least a $2^d$ dependence on $d$.
We study the complexity of computing the projection of an arbitrary $d$-polytope along $k$ orthogonal vectors for various input and output forms. We show that if $d$ and $k$ are part of the input (i.e. not a constant) and we are interested in output-
A biconvex polytope is a convex polytope that is also tropically convex. It is well known that every bounded cell of a tropical linear space is a biconvex polytope, but its converse has been a conjecture. We classify biconvex polytopes, and prove the
It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes: combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple