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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 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

We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…

Logic · Mathematics 2026-01-21 Meng-Che "Turbo" Ho , Martin Ritter , Luca San Mauro

The constraint satisfaction problem (CSP) can be formulated as a homomorphism problem between relational structures: given a structure $\mathcal{A}$, for any structure $\mathcal{X}$, whether there exists a homomorphism from $\mathcal{X}$ to…

Logic · Mathematics 2024-03-12 Azza Gaysin

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

An elegant characterization of the complexity of constraint satisfaction problems has emerged in the form of the the algebraic dichotomy conjecture of [BKJ00]. Roughly speaking, the characterization asserts that a CSP {\Lambda} is tractable…

Computational Complexity · Computer Science 2015-01-08 Jonah Brown-Cohen , Prasad Raghavendra

Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of…

Logic in Computer Science · Computer Science 2017-04-28 Catarina Carvalho , Barnaby Martin , Dmitriy Zhuk

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic…

Combinatorics · Mathematics 2016-07-22 Marcel Jackson , Tomasz Kowalski , Todd Niven

A ternary permutation constraint satisfaction problem (CSP) is specified by a subset Pi of the symmetric group S_3. An instance of such a problem consists of a set of variables V and a set of constraints C, where each constraint is an…

Computational Complexity · Computer Science 2014-10-10 Leo van Iersel , Steven Kelk , Nela Lekic , Simone Linz

The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP…

Computational Complexity · Computer Science 2016-05-03 Marcin Kozik , Joanna Ochremiak

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of…

Data Structures and Algorithms · Computer Science 2015-07-21 Robert Ganian , M. S. Ramanujan , Stefan Szeider

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem…

Logic in Computer Science · Computer Science 2010-03-04 Florent Madelaine , Barnaby Martin

In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the…

Computational Complexity · Computer Science 2019-04-10 Carlos Areces , Miguel Campercholi , Daniel Penazzi , Pablo Ventura

A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…

Logic · Mathematics 2013-08-30 Andre Nies

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

Every CSP(B) for a finite structure B is either in P or it is NP-complete but the proofs of the finite-domain CSP dichotomy by Andrei Bulatov and Dimitryi Zhuk not only show the computational complexity separation but also confirm the…

Logic in Computer Science · Computer Science 2024-02-27 Michal Wrona

We prove a complexity dichotomy theorem for all non-negative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms…

Computational Complexity · Computer Science 2010-12-30 Jin-Yi Cai , Xi Chen , Pinyan Lu

In this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension…

Computational Complexity · Computer Science 2015-04-29 Petr Kolman , Martin Koutecký

Literature on Constraint Satisfaction exhibits the definition of several structural properties that can be possessed by CSPs, like (in)consistency, substitutability or interchangeability. Current tools for constraint solving typically…

Artificial Intelligence · Computer Science 2014-01-16 Lucas Bordeaux , Marco Cadoli , Toni Mancini

A backdoor in a finite-domain CSP instance is a set of variables where each possible instantiation moves the instance into a polynomial-time solvable class. Backdoors have found many applications in artificial intelligence and elsewhere,…

Computational Complexity · Computer Science 2022-11-21 Peter Jonsson , Victor Lagerkvist , Sebastian Ordyniak