We consider a matching problem in a bipartite graph $G=(A\cup B,E)$ where nodes in $A$ are agents having preferences in partial order over their neighbors, while nodes in $B$ are objects without preferences. We propose a polynomial-time combinatorial algorithm based on LP duality that finds a maximum matching or assignment in $G$ that is popular among all maximum matchings, if there exists one. Our algorithm can also be used to achieve a trade-off between popularity and cardinality by imposing a penalty on unmatched nodes in $A$. We also provide an $O^*(|E|^k)$ algorithm that finds an assignment whose unpopularity margin is at most $k$; this algorithm is essentially optimal, since the problem is $\mathsf{NP}$-complete and $\mathsf{W}_l[1]$-hard with parameter $k$. We also prove that finding a popular assignment of minimum cost when each edge has an associated binary cost is $\mathsf{NP}$-hard, even if agents have strict preferences. By contrast, we propose a polynomial-time algorithm for the variant of the popular assignment problem with forced/forbidden edges. Finally, we present an application in the context of housing markets.

翻译：我们考虑二分图$G=(A\cup B,E)$中的匹配问题，其中$A$中的节点是具有偏序偏好的智能体，而$B$中的节点是无偏好的对象。我们提出一种基于LP对偶的多项式时间组合算法，该算法能在存在解的情况下找到$G中所有最大匹配中流行的最大匹配或指派。该算法还可通过对$A$中未匹配节点施加惩罚，实现流行度与基数之间的权衡。我们还提供一种复杂度为$O^*(|E|^k)$的算法，用于找到不流行度界限不超过$k$的指派；该算法本质上是最优的，因为该问题是$\mathsf{NP}$-完全的，且关于参数$k$是$\mathsf{W}_l[1]$-困难的。我们进一步证明，即使用户具有严格偏好，当每条边关联二元成本时，寻找最小成本流行指派问题是$\mathsf{NP}$-困难的。相比之下，我们针对带有强制/禁止边的流行指派问题变体提出多项式时间算法。最后，我们展示了该问题在住房市场背景下的应用。