发表机构
UNSW Sydney; HUN-REN SZTAKI; Budapest University of Technology and Economics; Kyushu University; CyberAgent, AI Lab（新南威尔士大学; 匈牙利科学院计算与系统科学研究所; 布达佩斯科技大学; 九州大学; CyberAgent AI实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对家庭与日托中心匹配中联合偏好与初始入学带来的挑战，分解稳定性为公平性与非浪费性，提出重新安置与贪心算法及公平保持非浪费算法，并刻画了相关计算复杂性。
AI 中文摘要
我们研究家庭与日托中心之间的双边匹配问题，其中家庭可能有多个孩子且具有联合偏好，并且一些孩子可能已经入托而寻求转学。我们的模型结合并推广了两个研究充分的匹配框架：夫妻匹配和带初始入学的学校选择。联合家庭偏好与初始入学之间的相互作用对理想结果的存在性和计算高效算法的设计提出了根本性挑战。由于稳定匹配不一定存在，我们将稳定性分解为个体理性、公平性和非浪费性，并从两个互补方向研究公平性与非浪费性之间的权衡。为了保持非浪费性，我们首先考虑基于主列表的方法，并表明它们在存在初始入学的情况下是不充分的。然后我们引入“重新安置”，通过允许流离失所的家庭返回其初始分配来保护他们，并开发了一种贪心改进算法，该算法确立了满足非浪费性以及基于重新安置的公平性的匹配的存在性。为了保持公平性，我们首先检查一种广义的截止方法，然后开发一种保持公平的非浪费算法，该算法保证公平性以及一种宽松的非浪费性概念。我们进一步刻画了这些解概念的计算复杂性。虽然若干公平性概念可以在多项式时间内验证，但判定是否存在满足这些概念的非浪费匹配是NP完全的。重新安置恢复了普遍存在性，但计算基于重新安置的结果是PLS难的，并且验证更强的基于优势的公平性概念是coNP完全的。
英文摘要
We study a two-sided matching problem between families and daycare centers in which a family may have multiple children with joint preferences, and some children may already be enrolled in daycare centers while seeking transfers. Our model combines and generalizes two well-studied matching frameworks: matching with couples and school choice with initial enrollments. The interaction between joint family preferences and initial enrollments creates fundamental challenges for the existence of desirable outcomes and the design of computationally efficient algorithms. Since stable matchings need not exist, we decompose stability into individual rationality, fairness, and non-wastefulness, and study the trade-off between fairness and non-wastefulness from two complementary directions. To preserve non-wastefulness, we first consider master-list-based approaches and show that they are insufficient in the presence of initial enrollments. We then introduce \emph{resettlement}, which protects displaced families by allowing them to return to their initial assignments, and develop a greedy improvement algorithm that establishes the existence of matchings satisfying non-wastefulness together with resettlement-based fairness. To preserve fairness, we first examine a generalized cutoff approach and then develop a fairness-preserving non-wasteful algorithm that guarantees fairness together with a relaxed notion of non-wastefulness. We further characterize the computational complexity of these solution concepts. While several fairness notions can be verified in polynomial time, deciding whether a non-wasteful matching satisfying them exists is NP-complete. Resettlement restores universal existence, but computing resettlement-based outcomes is PLS-hard, and verifying the stronger dominance-based fairness notion is coNP-complete.
