Download PDF by Stanislav Živný (auth.): The Complexity of Valued Constraint Satisfaction Problems

By Stanislav Živný (auth.)

ISBN-10: 3642339735

ISBN-13: 9783642339738

ISBN-10: 3642339743

ISBN-13: 9783642339745

The subject of this e-book is the subsequent optimisation challenge: given a suite of discrete variables and a collection of services, each one looking on a subset of the variables, minimise the sum of the features over all variables. This basic study challenge has been studied inside of a number of assorted contexts of discrete arithmetic, computing device technology and synthetic intelligence less than diversified names: Min-Sum difficulties, MAP inference in Markov random fields (MRFs) and conditional random fields (CRFs), Gibbs power minimisation, valued constraint pride difficulties (VCSPs), and, for two-state variables, pseudo-Boolean optimisation.

In this e-book the writer offers basic concepts for analysing the constitution of such capabilities and the computational complexity of the minimisation challenge, and he offers a accomplished checklist of tractable situations. in addition, he demonstrates that the so-called algebraic method of VCSPs can be utilized not just for the quest for tractable VCSPs, but in addition for different questions equivalent to discovering the bounds to the applicability of sure algorithmic techniques.

The booklet is appropriate for researchers drawn to equipment and effects from the world of constraint programming and discrete optimisation.

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3. 4. R2,1 Rd,1 Ff,1 Gf,1 R2,2 R2,3 = R2 . Rd,2 = Rd . Ff,2 = Ff . Gf,2 = Gf . We will then consider important subsets of these languages defined for totallyordered domains, containing the so-called max-closed cost functions, which are defined below. Recall that the function Max denotes the standard binary function which returns the larger of its two arguments. 2 A cost function φ is called max-closed if { 2, Max } ∈ fPol({φ}). 1 Equivalently, φ is max-closed if Max, Max ∈ Mul({φ}). 3 For every d ≥ 2 we define the following: • Rmax d,m denotes the set of all max-closed relations of arity at most m over an ordered def domain of size d, and Rmax = m≥0 Rmax d d,m .

5, the upper part corresponds to projections and the bottom part to operations. We relax the definition so that (in some cases) both parts can be arbitrary operations. Recall that a clone of operations, C, is a set of operations on some fixed set D that contains all projections and is closed under superposition. The k-ary operations in a clone C will be denoted by C (k) . 3 (Weighting) We define a k-ary weighting of a clone C to be a function ω : C (k) → Q such that ω(f ) < 0 only if f is a projection and ω(f ) = 0.

For instance, if f : D 2 → D, then f (t1 , t2 ) = f (t1 [1], t2 [1]), . . , f (t1 [k], t2 [k]) for any t1 , t2 ∈ D k . 6 (Polymorphism [101]) Let R be an m-ary relation over a finite set D and let f be a k-ary operation on D. Then f is a polymorphism of R if f (t1 , . . , tk ) ∈ R for all choices of t1 , . . , tk ∈ R. 6 We denote by ei(k) : D k → D the projection operation that returns its (k) ith argument, that is, ei (x1 , . . , xn ) = xi . 1 It follows readily from the definitions that any projection ei polymorphism of all relations.

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