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Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the emph{tolerant testing} of juntas. Given black-box access to a Boolean function $f:{pm1}^{n} to {pm1}$, we give a $poly(k, frac{1}{varepsilon})$ query algorithm that distinguishes between functions that are $gamma$-close to $k$-juntas and $(gamma+varepsilon)$-far from $k$-juntas, where $k = O(frac{k}{varepsilon^2})$. In the non-relaxed setting, we extend our ideas to give a $2^{tilde{O}(sqrt{k/varepsilon})}$ (adaptive) query algorithm that distinguishes between functions that are $gamma$-close to $k$-juntas and $(gamma+varepsilon)$-far from $k$-juntas. To the best of our knowledge, this is the first subexponential-in-$k$ query algorithm for approximating the distance of $f$ to being a $k$-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in $k$). Our techniques are Fourier analytical and make use of the notion of normalized influences that was introduced by Talagrand [AoP, 1994].
Suppose that we are given an arbitrary graph $G=(V, E)$ and know that each edge in $E$ is going to be realized independently with some probability $p$. The goal in the stochastic matching problem is to pick a sparse subgraph $Q$ of $G$ such that the
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain t
We develop a polynomial time $Omegaleft ( frac 1R log R right)$ approximate algorithm for Max 2CSP-$R$, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size $R$, and w
MAX CLIQUE problem (MCP) is an NPO problem, which asks to find the largest complete sub-graph in a graph $G, G = (V, E)$ (directed or undirected). MCP is well known to be $NP-Hard$ to approximate in polynomial time with an approximation ratio of $1 +
A number of recent works have studied algorithms for entrywise $ell_p$-low rank approximation, namely, algorithms which given an $n times d$ matrix $A$ (with $n geq d$), output a rank-$k$ matrix $B$ minimizing $|A-B|_p^p=sum_{i,j}|A_{i,j}-B_{i,j}|^p$