We introduce a "derandomized" analogue of graph squaring. This
operation increases the connectivity of the graph (as measured by the
second eigenvalue) almost as well as squaring the graph does, yet only
increases the degree of the graph by a constant factor, instead of
squaring the degree.

One application of ... more >>>

We give two new characterizations of ($\F_2$-linear) locally testable error-correcting codes in terms of Cayley graphs over $\F_2^h$:

\begin{enumerate}
\item A locally testable code is equivalent to a Cayley graph over $\F_2^h$ whose set of generators is significantly larger than $h$ and has no short linear dependencies, but yields a ... more >>>

Higher order random walks (HD-walks) on high dimensional expanders have played a crucial role in a number of recent breakthroughs in theoretical computer science, perhaps most famously in the recent resolution of the Mihail-Vazirani conjecture (Anari et al. STOC 2019), which focuses on HD-walks on one-sided local-spectral expanders. In this ... more >>>

A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ... more >>>

The classic Impagliazzo--Nisan--Wigderson (INW) psesudorandom generator (PRG) (STOC `94) for space-bounded computation uses a seed of length $O(\log n \cdot \log(nwd/\varepsilon))$ to fool ordered branching programs of length $n$, width $w$, and alphabet size $d$ to within error $\varepsilon$. A series of works have shown that the analysis of the ... more >>>

Dinitz, Schapira, and Valadarsky (Algorithmica 2017) introduced the intriguing notion of expanding expanders -- a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only ... more >>>

Random walks on expanders are a powerful tool which found applications in many areas of theoretical computer science, and beyond. However, they come with an inherent cost -- the spectral expansion of the corresponding power graph deteriorates at a rate that is exponential in the length of the walk. As ... more >>>

The Zig-Zag product of two graphs, $Z = G \circ H$, was introduced in the seminal work of Reingold, Vadhan, and Wigderson (Ann. of Math. 2002) and has since become a pivotal tool in theoretical computer science. The classical bound, which is used throughout, states that the spectral expansion of ... more >>>