Let C, W ⊆ Z. If C + W = Z, then the set C is called an additive complement to W in Z. If no proper subset of C is an additive complement to W , then C is called a minimal additive complement. Let X ⊆ N. If there exists a positive integer T such that x+T ∈ X for all sufficiently large integers x ∈ X, then we call X eventually periodic.In this paper, we study the existence of a minimal complement to W when W is eventually periodic or not. This partially answers a problem of Nathanson.