Abstract
Constraint satisfaction problems present a general framework for studying a large class of algorithmic problems such as satisfaction of Boolean formulas, solving systems of equations over finite fields, graph colourings, as well as various applied problems in artificial intelligence (scheduling, allocation of cell phone frequencies, among others.) CSP (Constraint Satisfaction Problems) bring together graph theory, complexity theory and universal algebra. It is a well known result, due to Feder and Vardi, that any constraint satisfaction problem over a finite relational structure can be reduced to the homomorphism problem for a finite oriented graph. Until recently, it was not known whether this reduction preserves the type of the algorithm which solves the original constraint satisfaction problem, so that the same algorithm solves the corresponding digraph homomorphism problem. We look at how a recent construction due to Bulin, Deli´c, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs.
Highlights
The central topic of this thesis are Constraint Satisfaction Problems and their reduction to digraph homomorphisms
We look at how a recent construction due to Bulin, Delic, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs
PROBLEMS several aspects in which it can be viewed using definability in Datalog, an algebraic problem asking whether a certain structure admits particular polymorphisms or, in the context of digraphs, whether particular intuitive algorithms are adequate to solve the CSP in question
Summary
The central topic of this thesis are Constraint Satisfaction Problems and their reduction to digraph homomorphisms. A paper by Feder and Vardi [7], resulted in a famous conjecture asserting that any constraint satisfaction problem over a finite relational structure is either tractable (solvable in polynomial runtime) or intractable (NP-complete). Feder and Vardi showed in the same exposition that every constraint satisfaction problem over a finite structure can be reduced to the homomorphism problem for a finite balanced digraph. The main result of this thesis is to give an alternative proof of this result using the Datalog approach to prove bounded width This is in contrast to a purely algebraic approach used in [5] and [1], to give a similar proof for Feder and Vardi construction
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