We show that for $k \geq 5$, the PPSZ algorithm for $k$-SAT runs exponentially faster if there is an exponential number of satisfying assignments. More precisely, we show that for every $k\geq 5$, there is a strictly increasing function $f: [0,1] \rightarrow \mathbb{R}$ with $f(0) = 0$ that has the ... more >>>

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work ... more >>>

Let $C$ be a depth-3 $\Sigma\Pi\Sigma$ arithmetic circuit of size $s$,
computing a polynomial $f \in \mathbb{F}[x_1,\ldots, x_n]$ (where $\mathbb{F}$ = $\mathbb{Q}$ or
$\mathbb{C}$) with fan-in of product gates bounded by $d$. We give a
deterministic time $2^d \text{poly}(n,s)$ polynomial identity testing
algorithm to check whether $f \equiv 0$ or ... more >>>