Cohen, Peri and Ta-Shma (STOC'21) considered the following question: Assume the vertices of an expander graph are labelled by $\pm 1$. What "test" functions $f : \{\pm 1\}^t \to \{\pm1 \}$ can or cannot distinguish $t$ independent samples from those obtained by a random walk? [CPTS'21] considered only balanced labelling, ... more >>>

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$.
This yields a nearly tight ... more >>>

We prove tight size bounds on monotone switching networks for the NP-complete problem of
$k$-clique, and for an explicit monotone problem by analyzing a pyramid structure of height $h$ for
the P-complete problem of generation. This gives alternative proofs of the separations of m-NC
from m-P and of m-NC$^i$ from ... more >>>

We prove that any monotone switching network solving directed connectivity on $N$ vertices must have size $N^{\Omega(\log N)}$

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