Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios

ブラウニアン モーター ノ 1ジゲン 2ジゲン ドウリキガク オヨビ カクリツ ブンプ

京都大学0048新制・課程博士博士(情報学)甲第11958号情博第182号新制||情||40(附属図書館)23747UT51-2006-B137京都大学大学院情報学研究科数理工学専攻(主査)教授 宗像 豊哲, 教授 岩井 敏洋, 助教授 五十嵐 顕人学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDA