A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin-Courant rule for preferences single-peaked on trees. We show that the egalitarian version of this problem admits a polynomial-time algorithm. For the utilitarian version, we prove that winner determination remains NP-hard, even for the Borda scoring function; however, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We also consider several other optimization criteria for trees: for some we obtain polynomial-time algorithms, while for others we show NP-hardness results.

中文翻译：

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树上的单峰偏好：多赢者选举和结构性结果

如果候选集可以配备树结构，使得每个选民的偏好沿着树中的所有路径从他们的顶级候选人开始递减，那么偏好配置文件在树上是单峰的。这个概念是由 Demange (1982) 引入的，随后 Trick (1989) 描述了一种有效的算法，用于确定给定的配置文件是否在树上是单峰的。我们研究了在张伯伦-库朗规则的几种变体下，多赢者选举的复杂性，用于偏好在树上单峰化。我们证明了这个问题的平等版本承认多项式时间算法。对于功利主义版本，我们证明获胜者确定仍然是 NP-hard，即使对于 Borda 评分函数也是如此；然而，如果底层树的叶子数或内部顶点数受常数限制，则可以在多项式时间内找到获胜委员会。为了从这些积极的结果中受益，我们需要一个程序来确定给定的配置文件是否在具有额外所需属性（例如，少量叶子）的树上是单峰的。为了应对这一挑战，我们开发了一种结构方法，使我们能够紧凑地表示给定轮廓是单峰的所有树。我们展示了如何使用这种表示有效地为给定的配置文件找到最佳树，以用于我们的获胜者确定算法：给定一个配置文件，我们可以有效地找到具有最少叶子数的树，或者在轮廓为单峰的树中具有最少内部顶点数的树。我们还考虑了树的其他几个优化标准：对于某些我们获得多项式时间算法，而对于其他我们显示 NP-hardness 结果。