A short story of the CSP dichotomy conjecture

Abstract

It has been observed long time ago that ‘natural’ computational problems tend to be complete in ‘natural’ complexity classes such as NL, P, NP, or PSPACE. Although Ladner in 1975 proved that if $\mathrm{P}\neq \mathrm{NP}$ then there are infinitely many complexity classes between them, all the examples of such intermediate problems are based on diagonalization constructions and are very artificial. Since the seminal work by Feder and Vardi [8] this phenomenon is known as complexity dichotomy (for P and NP), see also Valiant's work [14] in the context of counting problems. Concerted efforts have been made to make this observation more precise, and since the concept of a ‘natural’ problem is somewhat ambiguous, a possible research direction is to pursue dichotomy results for wide classes of problems. The Constraint Satisfaction problem (CSP) is one of such classes.