Explanipedia

Satisfiability of Commutative vs. Non-Commutative CSPs Open

Andreǐ A. Bulatov, Stanislav Živný · 2025

The Mermin-Peres magic square is a celebrated example of a system of Boolean linear equations that is not (classically) satisfiable but is satisfiable via linear operators on a Hilbert space of dimension four. A natural question is then, f…

Modular Counting CSP: Reductions and Algorithms Open

Amirhossein Kazeminia, Andreǐ A. Bulatov · 2025

The Constraint Satisfaction Problem (CSP) is ubiquitous in various areas of mathematics and computer science. Many of its variations have been studied including the Counting CSP, where the goal is to find the number of solutions to a CSP i…

Modular Counting Csp: Reductions and Algorithms Open

Andreǐ A. Bulatov, Amirhossein Kazeminia · 2025

Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras Open

Libor Barto, Zarathustra Brady, Andreǐ A. Bulatov, Marcin Kozik, Dmitriy Zhuk · 2024

This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to st…

The Ideal Membership Problem and Abelian Groups Open

Andreǐ A. Bulatov, Akbar Rafiey · 2022

Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a …

Minimal Taylor Algebras as a Common Framework for the Three Algebraic Approaches to the CSP Open

Libor Barto, Zarathustra Brady, Andreǐ A. Bulatov, Marcin Kozik, Dmitriy Zhuk · 2021

This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to st…

Complexity classification of counting graph homomorphisms modulo a prime number Open

Andreǐ A. Bulatov, Amirhossein Kazeminia · 2021

Counting graph homomorphisms and its generalizations such as the Counting Constraint Satisfaction Problem (CSP), its variations, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the…

Algebra of Modular Systems: Containment and Equivalence Open

Andreǐ A. Bulatov, Eugenia Ternovska · 2021

The Algebra of Modular System is a KR formalism that allows for combinations of modules written in multiple languages. Informally, a module represents a piece of knowledge. It can be given by a knowledge base, be an agent, an ASP, ILP, CP …

Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras Open

Libor Barto, Zarathustra Brady, Andreǐ A. Bulatov, Marcin Kozik, Dmitriy Zhuk · 2021

This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to st…

Symmetries and Complexity (Invited Talk) Open

Andreǐ A. Bulatov · 2021

The Constraint Satisfaction Problem (CSP) and a number of problems related to it have seen major advances during the past three decades. In many cases the leading driving force that made these advances possible has been the so-called algeb…

Dismantlability, connectedness, and mixing in relational structures Open

Raimundo Briceño, Andreǐ A. Bulatov, Víctor Dalmau, Benoît Larose · 2020

Separation of congruence intervals and implications Open

Andreǐ A. Bulatov · 2020

The Constraint Satisfaction Problem (CSP) has been intensively studied in many areas of computer science and mathematics. The approach to the CSP based on tools from universal algebra turned out to be the most successful one to study the c…

Graphs of relational structures: restricted types Open

Andreǐ A. Bulatov · 2020

The algebraic approach to the Constraint Satisfaction Problem (CSP) uses high order symmetries of relational structures -- polymorphisms -- to study the complexity of the CSP. In this paper we further develop one of the methods the algebra…

Local structure of idempotent algebras II Open

Andreǐ A. Bulatov · 2020

In this paper we continue the study of edge-colored graphs associated with finite idempotent algebras initiated in arXiv:2006.09599. We prove stronger connectivity properties of such graphs that will allows us to demonstrate several useful…

Local structure of idempotent algebras I Open

Andreǐ A. Bulatov · 2020

We refine and advance the study of the local structure of idempotent finite algebras started in [A.Bulatov, The Graph of a Relational Structure and Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like structure on an ar…

Approximate counting CSP seen from the other side Open

Andreǐ A. Bulatov, Stanislav Živný · 2019

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP($\mathcal{C}$,-), in which the goal is, given a relational structure $\mathbf{A}$ from a class $\mathcal{C}$ of structures and an ar…

Dismantlability, Connectedness, and Mixing in Relational Structures Open

Raimundo Brice no, Andreǐ A. Bulatov, Víctor Dalmau, Benot Larose · 2019

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a cruc…

Counting Homomorphisms Modulo a Prime Number Open

Amirhossein Kazeminia, Andreǐ A. Bulatov · 2019

Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(…

Counting homomorphisms in plain exponential time Open

Amineh Dadsetan, Andreǐ A. Bulatov · 2018

In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there…

A Dichotomy Theorem for Nonuniform CSPs Open

Andreǐ A. Bulatov · 2017

In this paper we prove the Dichotomy Conjecture on the complexity of nonuniform constraint satisfaction problems posed by Feder and Vardi.

Preface Open

Andreǐ A. Bulatov, Edward Hirsch, Jean-Éric Pin · 2017

Constraint Satisfaction Problems over semilattice block Mal'tsev algebras Open

Andreǐ A. Bulatov · 2017

There are two well known types of algorithms for solving CSPs: local propagation and generating a basis of the solution space. For several years the focus of the CSP research has been on `hybrid' algorithms that somehow combine the two app…

The subpower membership problem for semigroups Open

Andreǐ A. Bulatov, Marcin Kozik, Peter Mayr, Markus Steindl · 2016

Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [For…

Lower Bounds on Words Separation: Are There Short Identities in Transformation Semigroups? Open

Andreǐ A. Bulatov, Olga M. Karpova, Arseny M. Shur, Konstantin Startsev · 2016

The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n…

Lower Bounds on Words Separation: Are There Short Identities in\n Transformation Semigroups? Open

Andreǐ A. Bulatov, О. В. Карпова, Arseny M. Shur, Konstantin Startsev · 2016

The words separation problem, originally formulated by Goralcik and Koubek\n(1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for\nany two words of length $\\le n$ there is a deterministic finite automaton with\n$S…

Preface Open

Andreǐ A. Bulatov, Stephan Kreutzer · 2016

Graphs of finite algebras, edges, and connectivity Open

Andreǐ A. Bulatov · 2016

We refine and advance the study of the local structure of idempotent finite algebras started in [A.Bulatov, The Graph of a Relational Structure and Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like structure on an ar…

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