Combinatorial Proof that Subprojective Constraint Satisfaction Problems are NP-Complete

We introduce a new general polynomial-time constructionthe fibre constructionwhich reduces any constraint satisfaction problem CSP(H) to the constraint satisfaction problem CSP(P), where P is any subprojective relational system. As a consequence we get a new proof (not using universal algebra) that CSP(P) is NP -complete for any subprojective (and thus also projective) relational system. The fibre construction allows us to prove the NP-completeness part of the conjectured Dichotomy Classification of CSPs, previously obtained by algebraic methods. We show that this conjectured Dichotomy Classification is equivalent to the dichotomy of whether or not the template is subprojective. This approach is flexible enough to yield NP-completeness of coloring problems with large girth and bounded degree restrictions thus reducing the Feder-Hell-Huang and KostočkaNešeťril-Smoĺıková problems to the Dichotomy Classification of coloring problems.