Avi Wigderson, David Xiao

In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter, in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is ... more >>>

Avi Wigderson, David Xiao

Ahlswede and Winter introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan). As a consequence, we derandomize a construction of Alon ... more >>>

Stasys Jukna, Georg Schnitger

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained

by setting all *-entries to constants 0 or 1. A system of semi-linear

equations over GF(2) has the form Mx=f(x), where M is a completion of

A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate ...
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Arnab Bhattacharyya, Elena Grigorescu, Prasad Raghavendra, Asaf Shapira

Call a function $f: \mathbb{F}_2^n \to \{0,1\}$ odd-cycle-free if there are no $x_1, \dots, x_k \in \mathbb{F}_2^n$ with $k$ an odd integer such that $f(x_1) = \cdots = f(x_k) = 1$ and $x_1 + \cdots + x_k = 0$. We show that one can distinguish odd-cycle-free functions from those $\epsilon$-far ... more >>>

Vikraman Arvind, Partha Mukhopadhyay, Prajakta Nimbhorkar, Yadu Vasudev

Let $G=\langle S\rangle$ be a solvable permutation group given as input by generating set $S$. I.e.\ $G$ is a solvable subgroup of the symmetric group $S_n$. We give a deterministic polynomial-time algorithm that computes an expanding generator set for $G$. More precisely, given a constant $\lambda <1$ we can compute ... more >>>