We introduce a single-winner perspective on voting on matchings, in which voters have preferences over possible matchings in a graph, and the goal is to select a single collectively desirable matching. Unlike in classical matching problems, voters in our model are not part of the graph; instead, they have preferences over the entire matching. In the resulting election, the candidate space consists of all feasible matchings, whose exponential size renders standard algorithms for identifying socially desirable outcomes computationally infeasible. We study whether the computational tractability of finding such outcomes can be regained by exploiting the matching structure of the candidate space. Specifically, we provide a complete complexity landscape for questions concerning the maximization of social welfare, the construction and verification of Pareto optimal outcomes, and the existence and verification of Condorcet winners under one affine and two approval-based utility models. Our results consist of a mix of algorithmic and intractability results, revealing sharp boundaries between tractable and intractable cases, with complexity jumps arising from subtle changes in the utility model or solution concept.

翻译：本文提出了一种基于单胜者视角的匹配投票模型，在该模型中，投票者对图中可能的匹配方案持有偏好，目标是从中选出一个集体期望的匹配结果。与经典匹配问题不同，本模型中的投票者并非图的组成部分，而是对整个匹配方案具有偏好。在此类选举中，候选空间由所有可行匹配构成，其指数级规模使得识别社会期望结果的标准算法在计算上不可行。我们研究能否通过利用候选空间的匹配结构，重新获得求解此类结果的计算可操作性。具体而言，我们针对一个仿射效用模型和两个基于认可的效用模型，围绕社会福利最大化、帕累托最优结果的构建与验证、以及孔多塞胜者的存在性判定与验证等问题，给出了完整的计算复杂性图谱。我们的研究结果包含算法设计与不可行性证明，揭示了可处理情形与难处理情形之间的清晰界限，其中复杂性的突变源于效用模型或解概念的细微变化。