NP-completeness of generalized multi-Skolem sequences

A Skolem sequence is a sequence a1, a2, . . . , a2n (where ai ∈ A = {1, . . . , n}), each ai occurs exactly twice in the sequence and the two occurrences are exactly ai positions apart. A set A 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} are Skolem sets was solved by Thoralf Skolem [6] 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 NP-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.