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In this paper we study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language {\Gamma} consisting of finitary functions on a fixed finite domain. An instance of VCSP is given…

Computational Complexity · Computer Science 2018-03-22 Victor Dalmau , Andrei Krokhin , Rajsekar Manokaran

In the Boolean maximum constraint satisfaction problem - Max CSP$(\Gamma)$ - one is given a collection of weighted applications of constraints from a finite constraint language $\Gamma$, over a common set of variables, and the goal is to…

Computational Complexity · Computer Science 2020-02-11 Bart M. P. Jansen , Michał Włodarczyk

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 complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 20 years. A new version of the CSP, the promise CSP (PCSP) has recently been proposed, motivated by open questions about…

Computational Complexity · Computer Science 2021-07-19 Libor Barto , Jakub Bulín , Andrei Krokhin , Jakub Opršal

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…

Computational Complexity · Computer Science 2024-08-07 Libor Barto , Zarathustra Brady , Andrei Bulatov , Marcin Kozik , Dmitriy Zhuk

The constraint satisfaction problem (CSP) has important applications in computer science and AI. In particular, infinite-domain CSPs have been intensively used in subareas of AI such as spatio-temporal reasoning. Since constraint…

Artificial Intelligence · Computer Science 2021-07-06 Konrad K. Dabrowski , Peter Jonsson , Sebastian Ordyniak , George Osipov

The tractability conjecture for finite domain Constraint Satisfaction Problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise technical sense, from the 3-SAT problem;…

Logic in Computer Science · Computer Science 2021-01-12 Libor Barto , Michael Pinsker

The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NP-hard in general, considerable effort has been spent on identifying tractable subclasses. The main…

Artificial Intelligence · Computer Science 2014-07-09 David A. Cohen , Martin C. Cooper , Páidí Creed , András Z. Salamon

The minimum unsatisfiability version of a constraint satisfaction problem (MinCSP) asks for an assignment where the number of unsatisfied constraints is minimum possible, or equivalently, asks for a minimum-size set of constraints whose…

Computational Complexity · Computer Science 2018-05-09 Édouard Bonnet , László Egri , Bingkai Lin , Dániel Marx

The constrained synchronization problem (CSP) asks for a synchronizing word of a given input automaton contained in a regular set of constraints. It could be viewed as a special case of synchronization of a discrete event system under…

Formal Languages and Automata Theory · Computer Science 2021-08-03 Stefan Hoffmann

An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite…

Computational Complexity · Computer Science 2017-02-14 Vladimir Kolmogorov , Andrei Krokhin , Michal Rolinek

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In…

Data Structures and Algorithms · Computer Science 2018-12-06 Víctor Dalmau , Marcin Kozik , Andrei Krokhin , Konstantin Makarychev , Yury Makarychev , Jakub Opršal

Let A be an idempotent algebra on a finite domain. By mediating between results of Chen and Zhuk, we argue that if A satisfies the polynomially generated powers property (PGP) and B is a constraint language invariant under A (that is, in…

Computational Complexity · Computer Science 2021-06-25 Catarina Carvalho , Florent Madelaine , Barnaby Martin , Dmitriy Zhuk

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

A wide range of problems can be modelled as constraint satisfaction problems (CSPs), that is, a set of constraints that must be satisfied simultaneously. Constraints can either be represented extensionally, by explicitly listing allowed…

Artificial Intelligence · Computer Science 2015-02-10 Evgenij Thorstensen

Constraint Satisfaction Problems (CSP) constitute a convenient way to capture many combinatorial problems. The general CSP is known to be NP-complete, but its complexity depends on a template, usually a set of relations, upon which they are…

Computational Complexity · Computer Science 2010-11-23 Florian Richoux

These are notes from a multi-year learning seminar on the algebraic approach to Constraint Satisfaction Problems (CSPs). The main topics covered are the theory of algebraic structures with few subpowers, the theory of absorbing subalgebras…

Rings and Algebras · Mathematics 2025-05-27 Zarathustra Brady

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions over a finite domain. An instance of the problem is specified by…

Computational Complexity · Computer Science 2010-08-10 Vladimir Kolmogorov , Stanislav Zivny

It has been shown that for a general-valued constraint language $\Gamma$ the following statements are equivalent: (1) any instance of $\operatorname{VCSP}(\Gamma)$ can be solved to optimality using a constant level of the Sherali-Adams LP…

Computational Complexity · Computer Science 2018-05-28 Johan Thapper , Stanislav Zivny

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions over a finite domain. An instance of the problem is specified by…

Computational Complexity · Computer Science 2010-08-25 Vladimir Kolmogorov