发表机构
University of California, Los Angeles; University of Texas at Austin（加州大学洛杉矶分校; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究存在不可分割夫妻的医院/住院医师问题中寻找联盟稳定分配的复杂性，基于麦克德米德和曼洛夫的想法，证明此问题是NP难的，结果适用于HRIC，还引入单位联盟稳定概念并证明NP难结果对其也适用。
AI 中文摘要
在课程分配的近期工作中，罗德里格斯和曼洛夫考虑了在四种稳定概念下寻找稳定分配的复杂性，包括两种联盟概念。在一种被称为配对规模稳定的情况中，他们表明稳定分配总是存在并给出了多项式时间算法来找到一个。在第二种被称为配对稳定的情况中，他们观察到麦克德米德和曼洛夫早期关于课程分配特殊情况（称为带规模的医院/住院医师问题（HRS））的NP难结果成立。在第三种被称为第一联盟稳定的情况中，他们通过从HRS归约证明找到稳定分配是NP难的。他们留下了在所谓联盟稳定下寻找稳定分配的复杂性问题未解决。基于麦克德米德和曼洛夫的想法，我们解决了罗德里格斯和曼洛夫的开放问题，表明为HRS找到联盟稳定分配是NP难的。实际上，我们的证明表明当医院容量和住院医师规模至多为2时该问题仍然是NP难的。因此，我们的NP难结果适用于HRS的特殊情况，即存在不可分割夫妻的医院/住院医师问题（HRIC）。最后，我们为HRS和课程分配引入了一种新颖且自然的联盟稳定概念，并表明我们的NP难结果扩展到了这个我们称为单位联盟稳定的概念。
英文摘要
In recent work on course allocation, Rodríguez and Manlove consider the complexity of finding a stable assignment under four notions of stability, including two coalitional notions. In one case, which they call pair-size stability, they show that a stable assignment always exists and they provide a polynomial-time algorithm to find one. In a second case, called pair stability, they observe that an earlier NP-hardness result of McDermid and Manlove holds for a special case of course allocation called Hospitals/Residents with Sizes ($\mbox{HRS}$). In a third case, called first-coalition stability, they use a reduction from $\mbox{HRS}$ to show it is NP-hard to find a stable assignment. They leave open the complexity of finding a stable assignment under so-called coalition stability. Building on ideas from McDermid and Manlove, we resolve the open problem of Rodríguez and Manlove by showing that it is NP-hard to find a coalition-stable assignment for $\mbox{HRS}$. Indeed, our proof shows that the problem remains NP-hard when the hospital capacities and resident sizes are at most two. Accordingly, our NP-hardness result applies to the special case of $\mbox{HRS}$ known as Hospitals/Residents with Inseparable Couples ($\mbox{HRIC}$). Finally, we introduce a novel and natural notion of coalitional stability for both $\mbox{HRS}$ and course allocation, and we show that our NP-hardness result extends to this notion, which we call unitwise-coalition stability.
CommentsConference version of this paper to appear in SAGT 2026
