Colourings of hypergraphs, permutation groups and CSP's

Summary

Abstract. We investigate the complexity of strong colouring problems of hypergraphs associated to groups of permutations initiated in [13]. By reformulating these as Constraint Satisfaction Problems (CSP's) we are able to exploit recent algebraic results to answer various questions posed by Haddad and Rödl [13]. In particular, we show that all known tractable cases are explained by the presence of a Mal'tsev operation, and that all known NP-complete cases correspond to algebras that admit no Taylor operation, thus confirming the dichotomy conjecture by Bulatov et al. [7, 8]. We classify completely the case where the group consists of all affine transformations on a ring of matrices over a commutative ring.

Introduction. We start by briefly describing the strong colouring problem for permutation groups that was investigated in [13] and [12]; this problem has its origins in the classification of maximal partial clones over a finite non-empty set [14, 15, 16]. We shall immediately reformulate this decision problem as a Constraint Satisfaction Problem (CSP) in order to exploit various universal algebraic tools to study its algorithmic complexity.