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We study Constraint Satisfaction Problems (CSPs) in an infinite context. We show that the dichotomy between easy and hard problems -- established already in the finite case -- presents itself as the strength of the corresponding De…

Logic · Mathematics 2024-10-30 Tamás Kátay , László Márton Tóth , Zoltán Vidnyánszky

The constraint satisfaction problem asks to decide if a set of constraints over a relational structure $\mathcal{A}$ is satisfiable (CSP$(\mathcal{A})$). We consider CSP$(\mathcal{A} \cup \mathcal{B})$ where $\mathcal{A}$ is a structure and…

Computational Complexity · Computer Science 2024-08-28 Peter Jonsson , Victor Lagerkvist , George Osipov

Feder-Vardi conjecture, which proposed that every finite-domain Constraint Satisfaction Problem (CSP) is either in P or it is NP-complete, has been solved independently by Bulatov and Zhuk almost ten years ago. Bodirsky-Pinsker conjecture…

Computational Complexity · Computer Science 2026-04-06 Leonid Dorochko , Michał Wrona

A famous result by Jeavons, Cohen, and Gyssens shows that every constraint satisfaction problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the…

Computational Complexity · Computer Science 2011-11-30 Manuel Bodirsky , Dugald Macpherson , Johan Thapper

We give a complexity dichotomy theorem for the counting Constraint Satisfaction Problem (#CSP in short) with complex weights. To this end, we give three conditions for its tractability. Let F be any finite set of complex-valued functions,…

Computational Complexity · Computer Science 2015-03-19 Jin-Yi Cai , Xi Chen

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…

Logic · Mathematics 2025-01-16 Andrei A. Bulatov

A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set B of Boolean functions. We consider the problem of determining whether two given constraint…

Computational Complexity · Computer Science 2007-05-23 E. Boehler , E. Hemaspaandra , Steffen Reith , Heribert Vollmer

The constraint satisfaction problem (CSP) of a first-order theory T is the computational problem of deciding whether a given conjunction of atomic formulas is satisfiable in some model of T. We study the computational complexity of CSP$(T_1…

Logic · Mathematics 2023-06-22 Manuel Bodirsky , Johannes Greiner , Jakub Rydval

In the Constraint Satisfaction Problem (CSP for short) the goal is to decide the existence of a homomorphism from a given relational structure $G$ to a given relational structure $H$. If the structure $H$ is fixed and $G$ is the only input,…

Logic in Computer Science · Computer Science 2025-10-14 Andrei A. Bulatov , Amirhossein Kazeminia

We study the complexity of the valued constraint satisfaction problem (VCSP) for every valued structure with the domain ${\mathbb Q}$ that is preserved by all order-preserving bijections. Such VCSPs will be called temporal, in analogy to…

Logic · Mathematics 2026-01-21 Manuel Bodirsky , Édouard Bonnet , Žaneta Semanišinová

Constraint Satisfaction Problem on finite sets is known to be NP-complete in general but certain restrictions on the constraint language can ensure tractability. It was proved that if a constraint language has a weak near unanimity…

Computational Complexity · Computer Science 2019-03-07 Dmitriy Zhuk

One of the central open problems to classify the computational complexity of finite-domain constraint satisfaction problems within P is to prove better algorithmic results for CSPs with a Maltsev polymorphism; we do not even know whether…

Rings and Algebras · Mathematics 2026-02-10 Manuel Bodirsky , Andrew Moorhead

The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two…

Logic in Computer Science · Computer Science 2015-07-01 Barnaby Martin , Manuel Bodirsky , Martin Hils

For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust…

Data Structures and Algorithms · Computer Science 2025-09-09 Joshua Brakensiek , Venkatesan Guruswami , Sai Sandeep

Constraint satisfaction (CSP) and structure isomorphism (SI) are among the most well-studied computational problems in Computer Science. While neither problem is thought to be in $\texttt{PTIME},$ much work is done on $\texttt{PTIME}$…

Logic in Computer Science · Computer Science 2022-07-01 Adam Ó Conghaile

What makes a computational problem easy (e.g., in P, that is, solvable in polynomial time) or hard (e.g., NP-hard)? This fundamental question now has a satisfactory answer for a quite broad class of computational problems, so called…

Computational Complexity · Computer Science 2019-09-12 Libor Barto

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in…

Computational Complexity · Computer Science 2015-05-19 Manuel Bodirsky , Michael Pinsker

Constraint satisfaction problems (or CSPs) have been extensively studied in, for instance, artificial intelligence, database theory, graph theory, and statistical physics. From a practical viewpoint, it is beneficial to approximately solve…

Computational Complexity · Computer Science 2012-10-17 Tomoyuki Yamakami

Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well-known problems can be cast as promise CSPs including approximate graph…

Computational Complexity · Computer Science 2018-07-16 Joshua Brakensiek , Venkatesan Guruswami

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…

Computational Complexity · Computer Science 2020-05-15 Andrei A. Bulatov , Stanislav Zivny