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An instance of Max CSP is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max…

Computational Complexity · Computer Science 2007-12-11 Peter Jonsson , Andrei Krokhin , Fredrik Kuivinen

We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in…

Computational Complexity · Computer Science 2009-07-23 Martin Dyer , Leslie Ann Goldberg , Mark Jerrum

We study the Constraint Satisfaction Problem CSP(A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete.

Computational Complexity · Computer Science 2018-07-04 Manuel Bodirsky , Barnaby Martin , Marcello Mamino , Antoine Mottet

A Constraint Satisfaction Problem (CSP) is a computational problem where we are given variables and constraints about them; the question is whether the variables can be assigned values such that all constraints are satisfied. We give an…

Logic in Computer Science · Computer Science 2022-04-01 Michael Pinsker

Recent results show that a constraint satisfaction problem (CSP) defined over rational numbers with their natural ordering has a solution if and only if it has a definable solution. The proof uses advanced results from topology and modern…

Logic in Computer Science · Computer Science 2020-03-31 Michał R. Przybyłek

A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over $(\Bbb Z,<)$. Our main result says that every distance CSP is…

Logic · Mathematics 2016-04-27 Manuel Bodirsky , Barnaby Martin , Antoine Mottet

A constraint satisfaction problem (CSP), $\textsf{Max-CSP}(\mathcal{F})$, is specified by a finite set of constraints $\mathcal{F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$…

Computational Complexity · Computer Science 2024-02-27 Chi-Ning Chou , Alexander Golovnev , Madhu Sudan , Santhoshini Velusamy

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan…

Computational Complexity · Computer Science 2015-10-27 Konrad W. Schwerdtfeger

We study parameterized Constraint Satisfaction Problem for infinite constraint languages. The parameters that we study are weight of the satisfying assignment, number of constraints, maximum number of occurrences of a variable in the…

Computational Complexity · Computer Science 2017-08-10 Ruhollah Majdoddin

Valued constraint satisfaction problems (VCSPs) constitute a large class of computational optimization problems. It was shown recently that, over finite domains, every VCSP is in P or NP-complete, depending on the admitted cost functions.…

Logic · Mathematics 2025-10-21 Manuel Bodirsky , Žaneta Semanišinová , Carsten Lutz

The Quantified Constraint Satisfaction Problem is the problem of evaluating a sentence with both quantifiers, over relations from some constraint language, with conjunction as the only connective. We show that for any constraint language on…

Computational Complexity · Computer Science 2024-10-22 Dmitriy Zhuk

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 Problem (CSP) is a framework for modeling and solving a variety of real-world problems. Once the problem is expressed as a finite set of constraints, the goal is to find the variables' values satisfying them. Even…

Discrete Mathematics · Computer Science 2019-05-23 Rachid Oucheikh , Ismail Berrada , Outman El Hichami

The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and…

Category Theory · Mathematics 2022-11-04 Soichiro Fujii , Yuni Iwamasa , Kei Kimura

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

We study the complexity of constraint satisfaction problems for templates $\Gamma$ that are first-order definable in $(\Bbb Z; succ)$, the integers with the successor relation. Assuming a widely believed conjecture from finite domain…

Computational Complexity · Computer Science 2016-04-27 Manuel Bodirsky , Victor Dalmau , Barnaby Martin , Antoine Mottet , Michael Pinsker

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels…

Computational Complexity · Computer Science 2020-05-15 Peter Fulla , Hannes Uppman , Stanislav Zivny

The \emph{Sandwich Problem} (SP) for a graph class $\calC$ is the following computational problem. The input is a pair of graphs $(V,E_1)$ and $(V,E_2)$ where $E_1\subseteq E_2$, and the task is to decide whether there is an edge set $E$…

Discrete Mathematics · Computer Science 2025-10-13 Manuel Bodirsky , Santiago Guzmán-Pro

In this survey we describe progress over the last decade or so in understanding the complexity of solving constraint satisfaction problems (CSPs) approximately in the streaming and sketching models of computation. After surveying some of…

Computational Complexity · Computer Science 2022-05-06 Madhu Sudan

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template…