We study matching problems in which agents form one side of a bipartite graph and have preferences over objects on the other side. A central solution concept in this setting is popularity: a matching is popular if it is a (weak) Condorcet winner, meaning that no other matching is preferred by a strict majority of agents. It is well known, however, that Condorcet winners need not exist. We therefore turn to a natural and prominent relaxation. A set of matchings is a Condorcet-winning set if, for every competing matching, a majority of agents prefers their favorite matching in the set over the competitor. The Condorcet dimension is the smallest cardinality of a Condorcet-winning set. Our main results reveal a connection between Condorcet-winning sets and Pareto optimality. We show that any Pareto-optimal set of two matchings is, in particular, a Condorcet-winning set. This implication continues to hold when we impose matroid constraints on the set of matched objects, and even when agents' valuations are given as partial orders. The existence picture, however, changes sharply with partial orders. While for weak orders a Pareto-optimal set of two matchings always exists, this is -- surprisingly -- not the case under partial orders. Consequently, although the Condorcet dimension for matchings is 2 under weak orders (even under matroid constraints), this guarantee fails for partial orders: we prove that the Condorcet dimension is $Θ(\sqrt{n})$, and rises further to $Θ(n)$ when matroid constraints are added. On the computational side, we show that, under partial orders, deciding whether there exists a Condorcet -- winning set of a given fixed size is NP-hard. The same holds for deciding the existence of a Pareto-optimal matching, which we believe to be of independent interest. Finally, we also show that the Condorcet dimension for a related problem on arborescences is also 2.

翻译：我们研究二分图一侧的智能体对另一侧对象具有偏好时的匹配问题。该场景下的核心解概念是流行性：若一个匹配是（弱）孔多塞胜者，即不存在其他匹配被严格多数的智能体更偏好，则该匹配是流行的。然而众所周知，孔多塞胜者未必存在。因此我们转向一种自然且重要的松弛概念：若对于任意竞争匹配，多数智能体更偏好集合中他们最喜爱的匹配而非竞争者，则该匹配集合构成孔多塞获胜集。孔多塞维度即最小孔多塞获胜集的基数。我们的主要结果揭示了孔多塞获胜集与帕累托最优性之间的关联。我们证明任意由两个匹配构成的帕累托最优集，特别是，必然构成孔多塞获胜集。这一结论在匹配对象集合施加拟阵约束时依然成立，甚至在智能体估值以偏序形式给出时也成立。然而，存在性图景在偏序条件下发生显著变化。对于弱序，由两个匹配构成的帕累托最优集总是存在，但令人惊讶的是，在偏序条件下这一结论不再成立。因此，尽管在弱序条件下（即使在拟阵约束下）匹配问题的孔多塞维度为2，这一保证在偏序条件下失效：我们证明此时孔多塞维度为$Θ(\sqrt{n})$，而添加拟阵约束后进一步上升至$Θ(n)$。在计算复杂性方面，我们证明在偏序条件下，判定是否存在给定固定规模的孔多塞获胜集是NP难的。判定帕累托最优匹配的存在性同样具有NP难度，我们认为这一结论具有独立的研究价值。最后，我们还证明了有向树相关问题的孔多塞维度同样为2。