Dimitris Fotakis, Paul Spirakis

In this work we use random walks on expanders in order to

relax the properties of hitting sets required for partially

derandomizing one-side error algorithms. Building on a well-known

probability amplification technique [AKS87,CW89,IZ89], we use

random walks on expander graphs of subexponential (in the

random bit complexity) size so as ...
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Eric Allender

It is a trivial observation that every decidable set has strings of length $n$ with Kolmogorov complexity $\log n + O(1)$ if it has any strings of length $n$ at all. Things become much more interesting when one asks whether a similar property holds when one

considers *resource-bounded* Kolmogorov complexity. ...
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Maurice Jansen, Rahul Santhanam

Suppose $f$ is a univariate polynomial of degree $r=r(n)$ that is computed by a size $n$ arithmetic circuit.

It is a basic fact of algebra that a nonzero univariate polynomial of degree $r$ can vanish on at most $r$ points. This implies that for checking whether $f$ is identically zero, ...
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Noga Alon, Gil Cohen

We introduce a class of polynomials, which we call \emph{subspace polynomials} and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of ... more >>>

Aditya Bhaskara, Devendra Desai, Srikanth Srinivasan

We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests, including symmetric functions and combinatorial rectangles. Generalizing results of Linial, Luby, Saks, and Zuckerman (Combinatorica 1997) ... more >>>

Chandan Saha, Bhargav Thankey

The orbit of an $n$-variate polynomial $f(\mathbf{x})$ over a field $\mathbb{F}$ is the set $\mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}$. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>>

Venkatesan Guruswami, Xin Lyu, Xiuhan Wang

In the range avoidance problem, the input is a multi-output Boolean circuit with more outputs than inputs, and the goal is to find a string outside its range (which is guaranteed to exist). We show that well-known explicit construction questions such as finding binary linear codes achieving the Gilbert-Varshamov bound ... more >>>