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.

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