A Proof of the CSP Dichotomy Conjecture

Zhuk, Dmitriy

doi:10.1145/3402029

Cited by 195 publications

(294 citation statements)

References 49 publications

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“…We illustrate this for weak near-unanimity polymorphisms, given their importance in the characterisation of tractable languages [3,41]. A binary CSP instance I has the k-ary polymorphism f :…”

Section: Augmented Patterns: Motivationmentioning

confidence: 99%

“…An interesting avenue of theoretical research on CSPs consists in the characterisation of tractable subproblems defined by placing a restriction on the type of constraints that can occur (the constraint language) and again it is known that it is possible to limit attention to languages of binary relations [5,10]. A major advance towards the recent characterisation of tractable constraint languages [3,41] was the algebraic approach based on the study of pointwise closure operations of constraint relations, known as polymorphisms, and the identities satisfied by these polymorphisms [1,4]. Of particular interest is the Galois connection between (sets of) polymorphisms and (sets of) relations [27].…”

Section: Introductionmentioning

confidence: 99%

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Galois Connections for Patterns: An Algebra of Labelled Graphs

Cohen

Cooper

Jeavons

et al. 2021

Lecture Notes in Computer Science

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A pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any $$k \ge 1$$ k ≥ 1 , instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of instances such that each instance has a weak near-unanimity polymorphism of arity k (thus covering all tractable language classes).We investigate the power of forbidding augmented patterns and discuss their potential for describing new tractable classes.

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Section: Augmented Patterns: Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Galois Connections for Patterns: An Algebra of Labelled Graphs

Cohen

Cooper

Jeavons

et al. 2021

Lecture Notes in Computer Science

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“…implies the existence of a k-edge term for some k). This constitutes a major extension of current research into the complexity of (idempotent) Mal'tsev Condition Satisfaction Problems (MCSPs) which have been naturally viewed as the meta-question in constraint satisfaction since the celebrated proof of the CSP dichotomy theorem [5,15] and which also find independent applications in universal algebra and beyond, as well as being of interest from a purely complexity-theoretic perspective.…”

Section: J P Rooneymentioning

confidence: 99%

“…Hence the while loop is executed at most |A| k -many times (the largest possible size of a subuniverse of A k ). More importantly, lines ( 11)- (15) themselves are executed at most |A| kmany times for each basic operation of the algebra A. Line (15) itself is executed in time O(R|A| n ) where R is the maximum arity of the basic operations of A.…”

Section: Subalgebras and Operation Tablesmentioning

confidence: 99%

Mal’tsev condition satisfaction problems for conditions which imply edge terms

Rooney¹

2021

Algebra Univers.

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This paper considers the complexity of the problem of determining whether a finite idempotent algebra generates a variety which satisfies various strong Mal'tsev conditions. In particular we show that if the strong Mal'tsev condition under consideration implies the existence of an edge term, then deciding whether a finite idempotent algebra generates a variety which satisfies the Mal'tsev condition is in the complexity class NP. This extends an earlier result of Kazda, Opršal, Valeriote and Zhuk concerning minority terms to a much broader class of conditions, notably including all nontrivial linear strong Mal'tsev conditions in which every equation relates a term to a variable. The result of Kazda et al. being an application of a result of Mayr's concerning subpower membership problems, we note that our new result may similarly be seen as an application of the results on subpower membership problems obtained by Bulatov, Mayr and Szendrei. We also show that for fixed k ≥ 2 there is a polynomial-time procedure which takes an idempotent algebra A and produces the operation table of a k-edge term operation of A if such a term exists or else correctly returns the answer that no such term exists, again extending an earlier result of Kazda et al.

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“…Following [22], CSPs can be formulated as the problem of deciding the existence of a homomorphism from a finite relational structure G to a target relational structure H, where G and H encode the variables and the values of the CSP. The complexity of this problem, especially the case when H is a fixed finite relational structure, has received a lot of attention, culminating with the proof of the Feder-Vardi conjecture [14,44]. In the present paper we focus as well on the case when H is finite, although our main focus is not algorithmic but rather structural.…”

Section: Introductionmentioning

confidence: 99%

Dismantlability, connectedness, and mixing in relational structures

Briceño

Bulatov

Dalmau

et al. 2021

Journal of Combinatorial Theory, Series B

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The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, statistical physics, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, topological properties of the solution set such as connectedness is related to the hardness of CSPs over random structures. In approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions or the free energy of spin systems. Additionally, in the decision CSPs, structural properties of the relational structures involved -like, for example, dismantlability -and their logical characterizations have been instrumental for determining the complexity and other properties of the problem.In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that the structural property of dismantlability of the target graph, the connectedness of the set of homomorphisms, good mixing properties of the corresponding spin system, and the uniqueness of Gibbs measure are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.

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A Proof of the CSP Dichotomy Conjecture

Cited by 195 publications

References 49 publications

Galois Connections for Patterns: An Algebra of Labelled Graphs

Galois Connections for Patterns: An Algebra of Labelled Graphs

Mal’tsev condition satisfaction problems for conditions which imply edge terms

Dismantlability, connectedness, and mixing in relational structures

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