The Dichotomy for Conservative Constraint Satisfaction Problems Revisited

Barto, Libor

doi:10.1109/lics.2011.25

Cited by 61 publications

(184 citation statements)

References 20 publications

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“…THEOREM 2.10 [BULATOV 2003;BARTO 2011]. The dichotomy conjecture for the conservative homomorphism satisfaction problems holds.…”

Section: Constraint Satisfaction Problemmentioning

confidence: 97%

“…However, after the adoption of an algebraic approach, some significant results have been obtained. The most recent developments include a dichotomy theorem for nonuniform CSP over sets of values with three elements [Bulatov 2006] and a dichotomy theorem for nonuniform conservative CSP [Bulatov 2003;Barto 2011], that is, nonuniform CSP over a constraint language containing all unary relations. The proofs of those results are highly complex.…”

Section: Constraint Satisfaction Problemmentioning

confidence: 99%

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Why Is It Hard to Obtain a Dichotomy for Consistent Query Answering?

Fontaine

2015

ACM Trans. Comput. Logic

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A database may for various reasons become inconsistent with respect to a given set of integrity constraints. In the late 1990s, the formal approach of consistent query answering was proposed in order to query such databases. Since then, a lot of efforts have been spent to classify the complexity of consistent query answering under various classes of constraints. It is known that for the most common constraints and queries, the problem is in CONP and might be CONP-hard, yet several relevant tractable classes have been identified. Additionally, the results that emerged suggested that given a set of key constraints and a conjunctive query, the problem of consistent query answering is either in PTIME or is CONP-complete. However, despite all the work, as of today this dichotomy remains a conjecture.The main contribution of this article is to explain why it appears so difficult to obtain a dichotomy result in the setting of consistent query answering. Namely, we prove that such a dichotomy with respect to common classes of constraints and queries is harder to achieve than a dichotomy for the constraint satisfaction problem, which is a famous open problem since the 1990s.

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“…THEOREM 2.10 [BULATOV 2003;BARTO 2011]. The dichotomy conjecture for the conservative homomorphism satisfaction problems holds.…”

Section: Constraint Satisfaction Problemmentioning

confidence: 97%

Section: Constraint Satisfaction Problemmentioning

confidence: 99%

Why Is It Hard to Obtain a Dichotomy for Consistent Query Answering?

Fontaine

2015

ACM Trans. Comput. Logic

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show abstract

“…languages containing all unary relations) [6,25,27], A binary relation over a domain D can be viewed as a graph D, R . In the case of languages containing a single binary symmetric relation R, the Hell-Nešetřil theorem shows that is tractable if R viewed as a graph is bipartite or contains a loop, and is NP-complete otherwise [101].…”

Section: Dichotomiesmentioning

confidence: 99%

Tractability in constraint satisfaction problems: a survey

Carbonnel

Cooper

2015

Constraints

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Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. What is tractability?The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [83], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [118]. A computational decision problem (such as the constraint satisfaction problem or CSP) consists of a generic instance (in the case of the CSP, a set of variables, their

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“…Barto, Kozik and Niven [7] generalized Hell and Nešetřil's classification to directed graphs with no sources or sinks. For so-called maximal constraint languages and conservative constraint languages classifications were obtained in [18] and [14] (see also [4]) respectively. It is known that to confirm the dichotomy conjecture it is sufficient to consider binary constraint languages [19,33].…”

Section: Conjecture 33 (The Dichotomy Conjecture) Let γ Be a Finitementioning

confidence: 99%

On Some Combinatorial Optimization Problems : Algorithms and Complexity

Uppman¹

2015

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This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems.A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs.The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case.Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest.In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of iv optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.This work has been supported in part by the National Graduate School in Computer Science (CUGS), Sweden. Populärvetenskaplig sammanfattningOptimering går ut på att hitta värden för ett antal variabler så att resultatet blir så bra som möjligt (enligt en given kostnadsfunktion). Vi studerar kombinatoriska optimeringsproblem, problem där man för varje variabel enbart har ett ändligt antal möjliga val. Sådana problem är i någon mening lätta; för att hitta en så bra lösning som möjligt kan man helt enkelt prova alla sätt att välja värden för variablerna. Tyvärr är det här inte någon metod som fungerar i praktiken...

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The Dichotomy for Conservative Constraint Satisfaction Problems Revisited

Cited by 61 publications

References 20 publications

Why Is It Hard to Obtain a Dichotomy for Consistent Query Answering?

Why Is It Hard to Obtain a Dichotomy for Consistent Query Answering?

Tractability in constraint satisfaction problems: a survey

On Some Combinatorial Optimization Problems : Algorithms and Complexity

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