  Justified representation (JR) is a standard notion of representation in
multiwinner approval voting. Not only does a JR committee always exist, but
previous work has also shown through experiments that the JR condition can
typically be fulfilled by groups of fewer than $k$ candidates. In this paper,
we study such groups -- known as $n/k$-justifying groups -- both theoretically
and empirically. First, we show that under the impartial culture model,
$n/k$-justifying groups of size less than $k/2$ are likely to exist, which
implies that the number of JR committees is usually large. We then present
efficient approximation algorithms that compute a small $n/k$-justifying group
for any given instance, and a polynomial-time exact algorithm when the instance
admits a tree representation. In addition, we demonstrate that small
$n/k$-justifying groups can often be useful for obtaining a gender-balanced JR
committee even though the problem is NP-hard.
