We investigate the computational complexity of a class of PCSPs beyond the most studied cases – approximation variants of satisfiability and graph coloring problems. We give an almost complete classification for the class of PCSPs of the form: given a 3-uniform hypergraph that has an admissible 2-coloring, find an admissible 3-coloring, where admissibility is given by a ternary symmetric relation. The only PCSP of this sort whose complexity is left open in this work is a natural hypergraph coloring problem, where admissibility is given by the relation “if two colors are equal, then the remaining one is higher.”

This is joint work with Libor Barto and Diego Battistelli.