Testing Assignments to Constraint Satisfaction Problems

Chen, Hubie; Valeriote, Matthew; Yoshida, Yuichi

doi:10.1109/focs.2016.63

Cited by 7 publications

(4 citation statements)

References 43 publications

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“…We remark that appropriate versions of polymorphisms have been used extensively in many CSP complexity/approximability classifications: standard polymorphisms for decision and counting CSPs, for approximating Min CSPs and testing solutions (in the sense of property testing) [6,7,20,21,[24][25][26]48], fractional polymorphisms for exact optimisation problems [41,47], α-approximate polymorphisms for approximating Max CSPs [17]. In all cases, the presence of nice enough polymorphisms (of appropriate kind) leads directly to efficient algorithms, while their absence leads to hardness.…”

Section: Relatedmentioning

confidence: 99%

Algebraic approach to promise constraint satisfaction

Bulín

Krokhin

Opršal

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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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 the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms -high-dimensional symmetries of solution spaces -to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases.The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this paper we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem, and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a "measure of symmetry" that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by improving the state-of-the-art in approximate graph colouring: we show that, for any k ≥ 3, it is NP-hard to find a (2k − 1)-colouring of a given k-colourable graph. CCS CONCEPTS• Theory of computation → Problems, reductions and completeness.

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Section: Relatedmentioning

confidence: 99%

Algebraic approach to promise constraint satisfaction

Bulín

Krokhin

Opršal

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…While previous work on testing assignments to the problems CSP(A) studied concrete templates A or restricted classes of structures, this article presents a comprehensive classification of the constant query complexity templates. The results in this paper were first announced in [15]. Before describing our characterization, we introduce the algebraic notion of a polymorphism which is key to the description and obtention of our results.…”

Section: Contributionsmentioning

confidence: 99%

Constant-Query Testability of Assignments to Constraint Satisfaction Problems

Chen¹,

Valeriote²,

Yoshida³

2019

SIAM J. Comput.

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For each finite relational structure A, let CSP(A) denote the CSP instances whose constraint relations are taken from A. The resulting family of problems CSP(A) has been considered heavily in a variety of computational contexts. In this article, we consider this family from the perspective of property testing: given a CSP instance and query access to an assignment, one wants to decide whether the assignment satisfies the instance, or is far from doing so. While previous work on this scenario studied concrete templates or restricted classes of structures, this article presents a comprehensive classification theorem. Our main contribution is a dichotomy theorem completely characterizing the finite structures A such that CSP(A) is constant-query testable: • If A has a majority polymorphism and a Maltsev polymorphism, then CSP(A) is constantquery testable with one-sided error. • Else, testing CSP(A) requires a super-constant number of queries. CCS Concepts: • Theory of computation → Constraint and logic programming; Sketching and sampling;

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“…Near unanimity operations have played an important role in the development of universal algebra and first appeared in the 1970's in the work of Baker and Pixley [1] and Huhn [15]. More recently they have been used in the study of the CSP [13,16] and related questions [2,12]. The main results of this paper can be expressed in terms of the CSP and also in algebraic terms and we start by presenting them from both perspectives.…”

Section: Introductionmentioning

confidence: 99%

Sensitive instances of the Constraint Satisfaction Problem

Barto¹,

Kozik²,

Tan³

et al. 2020

Preprint

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We investigate the impact of modifying the constraining relations of a Constraint Satisfaction Problem (CSP) instance, with a fixed template, on the set of solutions of the instance. More precisely we investigate sensitive instances: an instance of the CSP is called sensitive, if removing any tuple from any constraining relation invalidates some solution of the instance. Equivalently, one could require that every tuple from any one of its constraints extends to a solution of the instance.Clearly, any non-trivial template has instances which are not sensitive. Therefore we follow the direction proposed (in the context of strict width) by Feder and Vardi in [13] and require that only the instances produced by a local consistency checking algorithm are sensitive. In the language of the algebraic approach to the CSP we show that a finite idempotent algebra A has a k + 2 variable near unanimity term operation if and only if any instance that results from running the (k, k + 1)-consistency algorithm on an instance over A 2 is sensitive.A version of our result, without idempotency but with the sensitivity condition holding in a variety of algebras, settles a question posed by G. Bergman about systems of projections of algebras that arise from some subalgebra of a finite product of algebras.Our results hold for infinite (albeit in the case of A idempotent) algebras as well and exhibit a surprising similarity to the strict width k condition proposed by Feder and Vardi. Both conditions can be characterized by the existence of a near unanimity operation, but the arities of the operations differ by 1.

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Testing Assignments to Constraint Satisfaction Problems

Cited by 7 publications

References 43 publications

Algebraic approach to promise constraint satisfaction

Algebraic approach to promise constraint satisfaction

Constant-Query Testability of Assignments to Constraint Satisfaction Problems

Sensitive instances of the Constraint Satisfaction Problem

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