The efficient computation of large matchings with desirable guarantees is a crucial objective in market design. However, even in simple two-sided matching markets with weak ordinal preferences, finding a maximum-size stable matching is NP-hard. Alternatively, popular matchings can be of larger size, but their existence is not guaranteed. In this paper, we study a new definition of popularity with two-sided weak preferences, where agents are only indifferent between two matchings if they receive the same partner. We show that this alternative definition of popularity, which we call weak popularity, guarantees the existence of such matchings. Unfortunately, finding a maximum-size weakly popular matching turns out to be NP-hard even with one-sided ties. However, we provide a polynomial-time algorithm to find a weakly popular matching that has at least $\frac{3}{4}$ times the size of a maximum-size weakly popular matching. We complement our approximation results with an Integer Linear Programming formulation that solves the maximum-size weakly popular matching problem exactly. We evaluate our algorithms on both randomly generated and real-world instances. Our experiments demonstrate that weakly popular matchings can be significantly larger than stable matchings, often covering all agents. Furthermore, we show that our approximation algorithm performs nearly optimally in practice. Finally, we show that maximum-size weakly popular matchings can have very few blocking edges, suggesting that weak popularity offers a desirable trade-off between size and stability. We also study a model more general than weak popularity, where for each edge, we can specify for both agents the size of improvement the agent needs to vote in favor of a new matching. We show that even in this more general model, a so-called $γ$-popular matching always exists, and our approximation algorithm applies.

翻译：在市场设计中，高效计算具有理想保证的大规模匹配是一个关键目标。然而，即使在具有弱序偏好的简单双边匹配市场中，寻找最大规模的稳定匹配也是NP难问题。作为替代方案，流行匹配可能具有更大规模，但其存在性无法保证。本文研究了一种基于双边弱偏好的新流行性定义，其中仅当智能体获得相同匹配对象时，它们才会对两种匹配方案持无差异态度。我们证明这种称为弱流行性的替代定义能够保证此类匹配的存在性。遗憾的是，即使存在单边平局约束，寻找最大规模的弱流行匹配仍是NP难问题。但我们提出了一种多项式时间算法，能够找到规模至少达到最大弱流行匹配$\frac{3}{4}$的弱流行匹配。我们通过整数线性规划模型对近似结果进行补充，该模型可精确求解最大规模弱流行匹配问题。我们在随机生成和真实世界实例上评估了所提算法。实验表明，弱流行匹配的规模可显著超越稳定匹配，通常能覆盖所有智能体。此外，我们的近似算法在实际应用中表现出接近最优的性能。最后，我们证明最大规模弱流行匹配可能仅包含极少数阻塞边，这表明弱流行性在匹配规模与稳定性之间提供了理想的权衡。我们还研究了比弱流行性更广义的模型，其中可为每条边的两个智能体分别设定支持新匹配方案所需的最小改进阈值。研究表明，即使在此广义模型中，所谓$γ$-流行匹配始终存在，且我们的近似算法仍适用。