Noga Alon, Shai Gutner

Color Coding is an algorithmic technique for deciding efficiently

if a given input graph contains a path of a given length (or

another small subgraph of constant tree-width). Applications of the

method in computational biology motivate the study of similar

algorithms for counting the number of copies of a ...
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Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio, Emanuele Viola

We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps)/\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = \Omega(1/(\eps^2 \cdot \log(1/\eps))). Using standard constructions of k-wise ... more >>>

Scott Aaronson

In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth ... more >>>

Noga Alon, Shachar Lovett

A family of permutations in $S_n$ is $k$-wise independent if a uniform permutation chosen from the family maps any distinct $k$ elements to any distinct $k$ elements equally likely. Efficient constructions of $k$-wise independent permutations are known for $k=2$ and $k=3$, but are unknown for $k \ge 4$. In fact, ... more >>>

Avishay Tal

We show that $AC^0$ circuits of depth $d$ and size $m$ have at most $2^{-\Omega(k/(\log m)^{d-1})}$ of their Fourier mass at level $k$ or above. Our proof builds on a previous result by H{\aa}stad (SICOMP, 2014) who proved this bound for the special case $k=n$. Our result is tight up ... more >>>