The assignment problem with nearly Monge arrays and incompatible partner indices

In this paper we study the d-dimensional assignment problem in which entries of the cost array satisfy the Monge property, except for ∞-entries, which may violate it. We assume that the ∞-entries are incurred by incompatible partner indices and their number is bounded by an upper bound λ for each index. We show that the problem can be solved in linear time for fixed d and λ, and it becomes strongly NP-hard if d or λ is part of the input.