🤖 AI Summary
This paper investigates weak and super popularity in bipartite matching with ties in preferences, and their relationship to stability. Motivated by the practical trade-off between stability and popularity in real-world decision-making, we formally define *weakly popular matchings*, prove their universal existence, and design a polynomial-time algorithm achieving a 3/4-approximation to the maximum-size weakly popular matching—guaranteeing size at least 4/5 that of a maximum stable matching. We show this approximation ratio is tight and that computing a maximum weakly popular matching is NP-hard. We further introduce a generalized *γ-popular* model and define a new NP-hard variant: *super popular matchings*. Our results unify the theoretical frameworks of popularity and stability, providing more practically relevant matching criteria for preference profiles containing ties.
📝 Abstract
In this paper, we study a slightly different definition of popularity in bipartite graphs $G=(U,W,E)$ with two-sided preferences, when ties are present in the preference lists. This is motivated by the observation that if an agent $u$ is indifferent between his original partner $w$ in matching $M$ and his new partner $w'
e w$ in matching $N$, then he may probably still prefer to stay with his original partner, as change requires effort, so he votes for $M$ in this case, instead of being indifferent. We show that this alternative definition of popularity, which we call weak-popularity allows us to guarantee the existence of such a matching and also to find a weakly-popular matching in polynomial-time that has size at least $frac{3}{4}$ the size of the maximum weakly popular matching. We also show that this matching is at least $frac{4}{5}$ times the size of the maximum (weakly) stable matching, so may provide a more desirable solution than the current best (and tight under certain assumptions) $frac{2}{3}$-approximation for such a stable matching. We also show that unfortunately, finding a maximum size weakly popular matching is NP-hard, even with one-sided ties and that assuming some complexity theoretic assumptions, the $frac{3}{4}$-approximation bound is tight. Then, we study a more general model than weak-popularity, where for each edge, we can specify independently for both endpoints the size of improvement the endpoint needs to vote in favor of a new matching $N$. We show that even in this more general model, a so-called $gamma$-popular matching always exists and that the same positive results still hold. Finally, we define an other, stronger variant of popularity, called super-popularity, where even a weak improvement is enough to vote in favor of a new matching. We show that for this case, even the existence problem is NP-hard.