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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

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

Many natural decision problems can be formulated as constraint satisfaction problems for reducts $\mathbb{A}$ of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite…

Logic · Mathematics 2023-06-22 Manuel Bodirsky , Antoine Mottet

The quantified constraint satisfaction problem (QCSP) is the problem of deciding, given a structure and a first-order prenex sentence whose quantifier-free part is the conjunction of atoms, whether or not the sentence holds on the…

Logic in Computer Science · Computer Science 2015-03-20 Hubie Chen

Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…

Data Structures and Algorithms · Computer Science 2012-04-13 Vinícius G. P. de Sá , Guilherme D. da Fonseca , Raphael Machado , Celina M. H. de Figueiredo

Constraint Satisfaction Problems (CSPs, for short) make up a class of problems with applications in many areas of computer science. The first classification of these problems was given by Schaeffer who showed that every CSP over the domain…

Computational Complexity · Computer Science 2025-08-18 Dejan Delic , John Marcoux

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

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

The Constraint Satisfaction Problem (CSP) framework offers a simple and sound basis for representing and solving simple decision problems, without uncertainty. This paper is devoted to an extension of the CSP framework enabling us to deal…

Artificial Intelligence · Computer Science 2013-02-21 Helene Fargier , Jerome Lang , Roger Martin-Clouaire , Thomas Schiex

We investigate a correspondence between the complexity hierarchy of constraint satisfaction problems and a hierarchy of logical compactness hypotheses for finite relational structures. It seems that the harder a constraint satisfaction…

Logic · Mathematics 2023-06-22 Danny Rorabaugh , Claude Tardif , David Wehlau

The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in many…

Combinatorics · Mathematics 2023-03-06 Jan Bok , Richard Brewster , Pavol Hell , Nikola Jedličková , Arash Rafiey

We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a…

Logic in Computer Science · Computer Science 2021-12-02 Libor Barto , William DeMeo , Antoine Mottet

We study a model of constraint satisfaction problems geared towards instances with few variables but with domain of unbounded size (udCSP). Our model is inspired by recent work on FPT algorithms for MinCSP where frequently both upper and…

Data Structures and Algorithms · Computer Science 2025-08-25 Peter Jonsson , Victor Lagerkvist , Jorke M. de Vlas , Magnus Wahlström

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting…

Logic in Computer Science · Computer Science 2007-05-23 Hubie Chen

The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism $\mathbf{R}\to \mathbf{\Gamma}$ between two relational structures, where $\mathbf{R}$ is defined over a domain $V$ and $\mathbf{\Gamma}$ is defined over a…

Computational Complexity · Computer Science 2023-11-21 Rustem Takhanov

Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with $n$ variables and $m$ clauses, there is a value of $m = \Omega(n)$ beyond which the CSP will be…

Data Structures and Algorithms · Computer Science 2016-11-07 Prasad Raghavendra , Satish Rao , Tselil Schramm

The $\mathcal{H}$-coloring problem for undirected simple graphs is a computational problem from a huge class of the constraint satisfaction problems (CSP): an $\mathcal{H}$-coloring of a graph $\mathcal{G}$ is just a homomorphism from…

Logic · Mathematics 2020-10-07 Azza Gaysin

Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are computational problems where we are given a set of variables denoting values from a finite domain B and a set of constraints on the variables. A solution to such a…

Computational Complexity · Computer Science 2011-10-14 Walter Bach , Hang Zhou

We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden…

Computational Complexity · Computer Science 2007-06-27 Gabor Kun , Jaroslav Nesetril

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design…

Data Structures and Algorithms · Computer Science 2014-07-30 Amey Bhangale , Swastik Kopparty , Sushant Sachdeva