  A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is an additive
complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if
$W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an
arbitrary subset in a group, the existence of minimal complement(s) depends on
its structure. The dual problem asks that given such a set, if it is a minimal
complement to some subset. Additive complements have been studied in the
context of representations of integers since the time of Erd\H{o}s, Hanani,
Lorentz and others. The notion of minimal complements is due to Nathanson. We
study tightness property of complement pairs $(W,W')$ such that both $W$ and
$W'$ are minimal to each other. These are termed co-minimal pairs and we show
that any non-empty finite set in an arbitrary free abelian group belongs to
some co-minimal pair. We also study infinite sets forming co-minimal pairs. At
the other extreme, motivated by unbounded arithmetic progressions in the
integers, we look at sets which can never be a part of any minimal pair. This
leads to a discussion on co-minimality, subgroups, approximate subgroups and
asymptotic approximate subgroups of $G$.
