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.

中文翻译：

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在多赢家批准投票中证明团体

正当代表 (JR) 是多方批准投票中代表的标准概念。不仅 JR 委员会始终存在，而且之前的工作还通过实验表明，JR 条件通常可以由少于 $k$ 的候选人的组来满足。在本文中，我们从理论上和经验上研究了此类群体——称为 $n/k$ 正当化群体。首先，我们表明在公正文化模型下，可能存在规模小于 $k/2$ 的 $n/k$ 正当化团体，这意味着 JR 委员会的数量通常很大。然后，我们提出了有效的近似算法，为任何给定的实例计算一个小的 $n/k$-justifying group，以及当实例接受树表示时的多项式时间精确算法。此外，