By Nadia Creignou, Phokion G. Kolaitis, Heribert Vollmer

Nowadays constraint delight difficulties (CSPs) are ubiquitous in lots of various components of laptop technological know-how, from man made intelligence and database platforms to circuit layout, community optimization, and concept of programming languages. therefore, it is very important study and pinpoint the computational complexity of yes algorithmic initiatives on the topic of constraint pride. The complexity-theoretic result of those projects could have an immediate impression on, for example, the layout and processing of database question languages, or concepts in data-mining, or the layout and implementation of planners.

This cutting-edge survey comprises the papers that have been invited through the organizers after end of a world Dagstuhl-Seminar on Complexity of Constraints, held in Dagstuhl citadel, Germany, in October 2006. a few audio system have been solicited to jot down surveys providing the state-of-the-art of their distinctiveness. those contributions have been peer-reviewed via specialists within the box and revised earlier than they have been collated to the nine papers of this quantity. additionally, the quantity features a reprint of a survey via Kolaitis and Vardi at the logical method of constraint pride that first seemed in 'Finite version thought and its Applications', released via Springer in 2007.

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He introduced a new property, weak separability, that plays a crucial role in the parameterized complexity of problems. Boolean Constraint Satisfaction Problems: When Does Post’s Lattice Help? 26. A relation R is weakly separable if 1. whenever x1 and x2 are in R, if x1 ∧ x2 is in R, then so is x1 ∨ x2 , 2. whenever x1 < x2 < x3 are in R (where < refers to the coordinate-wise order) then so is x1 ⊕ x2 ⊕ x3 . All operations are taken coordinate-wise. Marx got the following classiﬁcation result. 27.

Let Ξ ⊆ X × Y be a relation between X and Y and deﬁne operators αΞ : P X → P Y and βΞ : P Y → P X by αΞ X := {y ∈ Y | (∀x ∈ X)Ξ(x, y)} βΞ Y := {x ∈ X | (∀y ∈ Y )Ξ(x, y)}. Then the pair αΞ – βΞ is a Galois connection between P X and P Y. 42 F. B¨ orner 2. C. between P X and P Y and deﬁne a relation Ξ ⊆ X × Y by Ξ := {(x, y) ∈ X × Y | x ∈ β{y}} ( = {(x, y) ∈ X × Y | y ∈ α{x}} ). Then αΞ = α and βΞ = β. C. is determined by a relation Ξ ⊆ X × Y between X and Y, and each such relation deﬁnes a Galois connection.

This problem is in the complexity class DP, the class of languages equal to an intersection of two languages, one from NP and the other from coNP. Unless the polynomial hierarchy collapses, DP is a strict superclass of NP and coNP. , without restrictions on the clauses) under usual polynomial-time many-one reductions. Then US is a (supposedly proper) subclass of DP, showing that while Unique-Sat ∈ DP it is most likely not complete in this class. In order to reduce a problem Unique-Sat(Γ ) to another one we need a parsimonious reduction that exactly preserves the number of solutions.