All reports by Author Vladimir Podolskii:

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TR19-002
| 31st December 2018
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Alexander Kulikov, Ivan Mikhailin, Andrey Mokhov, Vladimir Podolskii#### Complexity of Linear Operators

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TR18-174
| 19th October 2018
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Anastasiya Chistopolskaya, Vladimir Podolskii#### Parity Decision Tree Complexity is Greater Than Granularity

Revisions: 2

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TR17-184
| 29th November 2017
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Vladimir Podolskii, Alexander A. Sherstov#### Inner Product and Set Disjointness: Beyond Logarithmically Many Parties

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TR16-158
| 9th October 2016
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Alexander Kulikov, Vladimir Podolskii#### Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates

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TR13-021
| 5th February 2013
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Kristoffer Arnsfelt Hansen, Vladimir Podolskii#### Polynomial threshold functions and Boolean threshold circuits

Alexander Kulikov, Ivan Mikhailin, Andrey Mokhov, Vladimir Podolskii

Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator $Ax$?

As we observe in this paper, this problem contains ... more >>>

Anastasiya Chistopolskaya, Vladimir Podolskii

We prove a new lower bound on the parity decision tree complexity $D_{\oplus}(f)$ of a Boolean function $f$. Namely, granularity of the Boolean function $f$ is the smallest $k$ such that all Fourier coefficients of $f$ are integer multiples of $1/2^k$. We show that $D_{\oplus}(f)\geq k+1$.

This lower bound is ... more >>>

Vladimir Podolskii, Alexander A. Sherstov

A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f\colon(\{0,1\}^n)^{k}\to\{0,1\}$ with $k\gg\log n$ parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every $k\geq\log n$, showing in both cases that $\Theta(1+\lceil\log n\rceil/\log\lceil1+k/\log n\rceil)$ bits are ... more >>>

Alexander Kulikov, Vladimir Podolskii

We study the following computational problem: for which values of $k$, the majority of $n$ bits $\text{MAJ}_n$ can be computed with a depth two formula whose each gate computes a majority function of at most $k$ bits? The corresponding computational model is denoted by $\text{MAJ}_k \circ \text{MAJ}_k$. We observe that ... more >>>

Kristoffer Arnsfelt Hansen, Vladimir Podolskii

We study the complexity of computing Boolean functions on general

Boolean domains by polynomial threshold functions (PTFs). A typical

example of a general Boolean domain is $\{1,2\}^n$. We are mainly

interested in the length (the number of monomials) of PTFs, with

their degree and weight being of secondary interest. We ...
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