  Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be
a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its
terms can be ordered such that their product equals $1_G$ and
$\sum_{i=1}^{\ell}\frac{1}{\ord(g_i)}\le 1$. Let $\mathsf {ti}(G)$ be the
smallest integer $t$ such that every sequence $S$ over $G$ with $|S|\ge t$ has
a tiny product-one subsequence. The direct problem is to obtain the exact value
of $\mathsf {ti}(G)$, while the inverse problem is to characterize the
structure of long sequences over $G$ which have no tiny product-one
subsequences. In this paper, we consider the inverse problem for cyclic groups
and we also study both direct and inverse problems for dihedral groups and
dicyclic groups.
