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 $w'\ne w$ 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.

翻译：本文研究二分图$G=(U,W,E)$中带有双边偏好且偏好列表存在平局时的另一种流行性定义。这一研究源于以下观察：如果某主体$u$在匹配$M$中的原伙伴$w$与匹配$N$中的新伙伴$w'\ne w$之间无差异，由于改变需要付出努力，该主体可能仍倾向于维持原伙伴关系，因此在此情况下会投票支持$M$而非保持中立。我们证明这种被称为弱流行的替代定义能够保证此类匹配的存在性，并可在多项式时间内找到规模至少为最大弱流行匹配$\frac{3}{4}$的弱流行匹配。同时证明该匹配规模至少为最大（弱）稳定匹配的$\frac{4}{5}$倍，因此可能比当前最优（且在特定假设下紧的）$\frac{2}{3}$近似稳定匹配方案更具优势。此外我们证明，即使在单边平局情况下，寻找最大规模弱流行匹配仍是NP困难的，且在某些复杂度理论假设下，$\frac{3}{4}$近似界是紧的。随后我们研究比弱流行更通用的模型，其中可为每条边独立指定两端端点支持新匹配$N$所需的改进幅度。我们证明即使在此通用模型中，所谓的$\gamma$-流行匹配始终存在，且相同的积极结果仍然成立。最后，我们定义另一种更强的流行性变体——超流行，其中即使是微弱的改进也足以促使投赞成票。我们证明在此情况下，甚至存在性判定问题都是NP困难的。