NPNP-completeness of generalized multi-Skolem sequences

A Skolem sequence is a sequence a1,a2,…,a2na1,a2,…,a2n (where ai∈A={1,…,n})ai∈A={1,…,n}), each aiai occurs exactly twice in the sequence and the two occurrences are exactly aiai positions apart. A set AA that can be used to construct Skolem sequences is called a Skolem set. The existence question of deciding which sets of the form A={1,…,n}A={1,…,n} are Skolem sets was solved by Skolem [On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957) 57–68] in 1957. Many generalizations of Skolem sequences have been studied. In this paper we prove that the existence question for generalized multi-Skolem sequences is NPNP-complete. This can be seen as an upper bound on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question.