On a problem of Nathanson on non-minimal additive complements

Let $C$ and $W$ be two sets of integers. If $C+W=\mathbb{Z}$, then $C$ is called an additive complement to $W$. We further call $C$ a minimal additive complement to $W$ if no proper subset of $C$ is an additive complement to $W$. Answering a problem of Nathanson in part, we give sufficient conditions of $W$ which has no minimal additive complements. Our result also extends the prior result of Chen and Yang.