All reports by Author Vladimir Podolskii:

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TR24-025
| 13th February 2024
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Mason DiCicco, Vladimir Podolskii, Daniel Reichman#### Nearest Neighbor Complexity and Boolean Circuits

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TR23-157
| 31st October 2023
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Vladimir Podolskii, Dmitrii Sluch#### One-Way Communication Complexity of Partial XOR Functions

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TR23-153
| 19th October 2023
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Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, Vladimir Podolskii#### Towards Simpler Sorting Networks and Monotone Circuits for Majority

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TR22-116
| 17th August 2022
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Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, Vladimir Podolskii#### Constant-Depth Sorting Networks

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TR20-017
| 18th February 2020
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Alexander Kozachinskiy, Vladimir Podolskii#### Multiparty Karchmer-Wigderson Games and Threshold Circuits

Revisions: 1

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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

Mason DiCicco, Vladimir Podolskii, Daniel Reichman

A nearest neighbor representation of a Boolean function $f$ is a set of vectors (anchors) labeled by $0$ or $1$ such that $f(x) = 1$ if and only if the closest anchor to $x$ is labeled by $1$. This model was introduced by Hajnal, Liu, and TurĂ¡n (2022), who studied ... more >>>

Vladimir Podolskii, Dmitrii Sluch

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y) = f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known ... more >>>

Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, Vladimir Podolskii

In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity $2$ and the generalized setting with operations of arity $k$, where $k$ is a ... more >>>

Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, Vladimir Podolskii

In this paper, we address sorting networks that are constructed from comparators of arity $k > 2$. That is, in our setting the arity of the comparators -- or, in other words, the number of inputs that can be sorted at the unit cost -- is a parameter. We study ... more >>>

Alexander Kozachinskiy, Vladimir Podolskii

We suggest a generalization of Karchmer-Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone ... more >>>

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 ...
more >>>