Algebras and Algorithms

Abstract

This presents the necessary background from universal algebra to describe the connection between algebra and the constraint satisfaction problems (CSP) and then review the progress that has been made towards settling the Dichotomy Conjecture of Feder and Vardi. They conjecture that the subclass of the CSP parametrized by a given finite relational structure will either lie in the complexity class P or be NP-complete. Work on the Dichotomy Conjecture has led to some surprising and fundamental results about finite algebras and has motivated research on a number of fronts. This also focuses on several results that deal with algorithmic questions about finite algebras. A typical sort of problem, one that is of particular relevance to the CSP, is to determine the complexity of deciding if a given finite algebra has a term operation that satisfies some prescribed set of equations.