Given two nonempty subsets W, W^′⊆ G in an arbitrary abelian group G, the set W^′ is said to be an additive complement to W if W+ W^′= G and it is minimal if no proper subset of W^′ is a complement to W. The notion was introduced by Nathanson and previous works by him, Chen–Yang, Kiss–Sándor–Yang, etc. focussed on G= Z. In this article, we focus on the higher rank case. We introduce the notion of “spiked subsets” and give necessary and sufficient conditions for the existence of minimal complements for them. This provides an answer to a problem of Nathanson in several contexts.