All reports by Author Nikolay Vereshchagin:

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TR17-043
| 3rd March 2017
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Alexey Milovanov, Nikolay Vereshchagin#### Stochasticity in Algorithmic Statistics for Polynomial Time

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TR13-178
| 14th December 2013
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Nikolay Vereshchagin#### Randomized communication complexity of appropximating Kolmogorov complexity

Revisions: 2

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TR13-007
| 8th January 2013
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Bruno Bauwens, Anton Makhlin, Nikolay Vereshchagin, Marius Zimand#### Short lists with short programs in short time

Revisions: 1

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TR11-167
| 6th December 2011
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Nikolay Vereshchagin#### Improving on Gutfreund, Shaltiel, and Ta-Shma's paper "If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances''

Revisions: 1

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TR10-163
| 3rd November 2010
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Harry Buhrman, Leen Torenvliet, Falk Unger, Nikolay Vereshchagin#### Sparse Selfreducible Sets and Nonuniform Lower Bounds

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TR10-091
| 14th May 2010
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Nikolay Vereshchagin#### An Encoding Invariant Version of Polynomial Time
Computable Distributions

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TR10-090
| 14th May 2010
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Nikolay Vereshchagin#### {Algorithmic Minimal Sufficient Statistics: a New Definition

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TR06-024
| 18th February 2006
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Harry Burhman, Lance Fortnow, Michal Koucky, John Rogers, Nikolay Vereshchagin#### Inverting onto functions might not be hard

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TR05-095
| 26th August 2005
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Noga Alon, Ilan Newman, Alexander Shen, Gábor Tardos, Nikolay Vereshchagin#### Partitioning multi-dimensional sets in a small number of ``uniform'' parts

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TR04-081
| 9th September 2004
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Harry Buhrman, Lance Fortnow, Ilan Newman, Nikolay Vereshchagin#### Increasing Kolmogorov Complexity

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TR04-080
| 7th September 2004
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Lance Fortnow, Troy Lee, Nikolay Vereshchagin#### Kolmogorov Complexity with Error

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TR04-054
| 5th June 2004
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Andrej Muchnik, Alexander Shen, Nikolay Vereshchagin, Mikhail V. Vyugin#### Non-reducible descriptions for conditional Kolmogorov complexity

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TR04-030
| 9th March 2004
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Nikolay Vereshchagin#### Kolmogorov complexity of enumerating finite sets

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TR01-089
| 29th October 2001
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Andrej Muchnik, Nikolay Vereshchagin#### Logical operations and Kolmogorov complexity. II

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TR01-088
| 29th October 2001
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Alexander Shen, Nikolay Vereshchagin#### Logical operations and Kolmogorov complexity

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TR01-087
| 29th October 2001
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Bruno Durand, Alexander Shen, Nikolay Vereshchagin#### Descriptive complexity of computable sequences

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TR01-086
| 29th October 2001
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Nikolay Vereshchagin#### Kolmogorov Complexity Conditional to Large Integers

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TR01-083
| 29th October 2001
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Nikolay Vereshchagin#### An enumerable undecidable set with low prefix complexity: a simplified proof

Revisions: 1

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TR00-035
| 6th June 2000
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Nikolay Vereshchagin, Mikhail V. Vyugin#### Independent minimum length programs to translate between given strings

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TR00-026
| 11th April 2000
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Andrei Romashchenko, Alexander Shen, Nikolay Vereshchagin#### Combinatorial Interpretation of Kolmogorov Complexity

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TR99-014
| 30th May 1999
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Alexander Razborov, Nikolay Vereshchagin#### One Property of Cross-Intersecting Families

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TR97-054
| 17th November 1997
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Ran Raz, Gábor Tardos, Oleg Verbitsky, Nikolay Vereshchagin#### Arthur-Merlin Games in Boolean Decision Trees

Alexey Milovanov, Nikolay Vereshchagin

A fundamental notion in Algorithmic Statistics is that of a stochastic object, i.e., an object having a simple plausible explanation. Informally, a probability distribution is a plausible explanation for $x$ if it looks likely that $x$ was drawn at random with respect to that distribution. In this paper, we ... more >>>

Nikolay Vereshchagin

The paper [Harry Buhrman, Michal Koucky, Nikolay Vereshchagin. Randomized Individual Communication Complexity. IEEE Conference on Computational Complexity 2008: 321-331] considered communication complexity of the following problem. Alice has a binary string $x$ and Bob a binary string $y$, both of length $n$, and they want to compute or approximate

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Bruno Bauwens, Anton Makhlin, Nikolay Vereshchagin, Marius Zimand

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any universal machine, it is possible to compute in polynomial time on ... more >>>

Nikolay Vereshchagin

Assume that $NP\ne RP$. Gutfreund, Shaltiel, and Ta-Shma in [Computational Complexity 16(4):412-441 (2007)] have proved that for every randomized polynomial time decision algorithm $D$ for SAT there is a polynomial time samplable distribution such that $D$ errs with probability at least $1/6-\epsilon$ on a random formula chosen with respect to ... more >>>

Harry Buhrman, Leen Torenvliet, Falk Unger, Nikolay Vereshchagin

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in $EXP^{NP}$, or even ... more >>>

Nikolay Vereshchagin

When we represent a decision problem,like CIRCUIT-SAT, as a language over the binary alphabet,

we usually do not specify how to encode instances by binary strings.

This relies on the empirical observation that the truth of a statement of the form ``CIRCUIT-SAT belongs to a complexity class $C$''

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Nikolay Vereshchagin

We express some criticism about the definition of an algorithmic sufficient statistic and, in particular, of an algorithmic minimal sufficient statistic. We propose another definition, which has better properties.

more >>>Harry Burhman, Lance Fortnow, Michal Koucky, John Rogers, Nikolay Vereshchagin

The class TFNP, defined by Megiddo and Papadimitriou, consists of

multivalued functions with values that are polynomially verifiable

and guaranteed to exist. Do we have evidence that such functions are

hard, for example, if TFNP is computable in polynomial-time does

this imply the polynomial-time hierarchy collapses?

We give a relativized ... more >>>

Noga Alon, Ilan Newman, Alexander Shen, Gábor Tardos, Nikolay Vereshchagin

Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log |E|) subsets so the all the resulting bipartite graphcs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of ... more >>>

Harry Buhrman, Lance Fortnow, Ilan Newman, Nikolay Vereshchagin

How much do we have to change a string to increase its Kolmogorov complexity. We show that we can

increase the complexity of any non-random string of length n by flipping O(sqrt(n)) bits and some strings

require

Omega(sqrt(n)) bit flips. For a given m, we also give bounds for ...
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Lance Fortnow, Troy Lee, Nikolay Vereshchagin

We introduce the study of Kolmogorov complexity with error. For a

metric d, we define C_a(x) to be the length of a shortest

program p which prints a string y such that d(x,y) \le a. We

also study a conditional version of this measure C_{a,b}(x|y)

where the task is, given ...
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Andrej Muchnik, Alexander Shen, Nikolay Vereshchagin, Mikhail V. Vyugin

Let a program p on input a output b. We are looking for a

shorter program p' having the same property (p'(a)=b). In

addition, we want p' to be simple conditional to p (this

means that the conditional Kolmogorov complexity K(p'|p) is

negligible). In the present paper, we prove that ...
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Nikolay Vereshchagin

Solovay has proven that

the minimal length of a program enumerating a set A

is upper bounded by 3 times the absolute value of the

logarithm of the

probability that a random program will enumerate A.

It is unknown whether one can replace the constant

3 by a smaller constant.

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Andrej Muchnik, Nikolay Vereshchagin

We invistigate what is the minimal length of

a program mapping A to B and at the same time

mapping C to D (where A,B,C,D are binary strings). We prove that

it cannot be expressed

in terms of Kolmogorv complexity of A,B,C,D, their pairs (A,B), (A,C),

..., their ...
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Alexander Shen, Nikolay Vereshchagin

We define Kolmogorov complexity of a set of strings as the minimal

Kolmogorov complexity of its element. Consider three logical

operations on sets going back to Kolmogorov

and Kleene:

(A OR B) is the direct sum of A,B,

(A AND B) is the cartesian product of A,B,

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Bruno Durand, Alexander Shen, Nikolay Vereshchagin

We study different notions of descriptive complexity of

computable sequences. Our main result states that if for almost all

n the Kolmogorov complexity of the n-prefix of an infinite

binary sequence x conditional to n

is less than m then there is a program of length

m^2+O(1) that for ...
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Nikolay Vereshchagin

Assume that for almost all n Kolmogorov complexity

of a string x conditional to n is less than m.

We prove that in this case

there is a program of size m+O(1) that given any sufficiently large

n outputs x.

Nikolay Vereshchagin

We present a simplified proof of Solovay-Calude-Coles theorem

stating that there is an enumerable undecidable set with the

following property: prefix

complexity of its initial segment of length n is bounded by prefix

complexity of n (up to an additive constant).

Nikolay Vereshchagin, Mikhail V. Vyugin

A string $p$ is called a program to compute $y$ given $x$

if $U(p,x)=y$, where $U$ denotes universal programming language.

Kolmogorov complexity $K(y|x)$ of $y$ relative to $x$

is defined as minimum length of

a program to compute $y$ given $x$.

Let $K(x)$ denote $K(x|\text{empty string})$

(Kolmogorov complexity of $x$) ...
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Andrei Romashchenko, Alexander Shen, Nikolay Vereshchagin

The very first Kolmogorov's paper on algorithmic

information theory was entitled `Three approaches to the

definition of the quantity of information'. These three

approaches were called combinatorial, probabilistic and

algorithmic. Trying to establish formal connections

between combinatorial and algorithmic approaches, we

prove that any ...
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Alexander Razborov, Nikolay Vereshchagin

Assume that A, B are finite families of n-element sets.

We prove that there is an element that simultaneously

belongs to at least |A|/2n sets

in A and to at least |B|/2n sets in B. We use this result to prove

that for any inconsistent DNF's F,G with OR ...
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Ran Raz, Gábor Tardos, Oleg Verbitsky, Nikolay Vereshchagin

It is well known that probabilistic boolean decision trees

cannot be much more powerful than deterministic ones (N.~Nisan, SIAM

Journal on Computing, 20(6):999--1007, 1991). Motivated by a question

if randomization can significantly speed up a nondeterministic

computation via a boolean decision tree, we address structural

properties of Arthur-Merlin games ...
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