Hospitals/Residents with Inseparable Couples: Finding a Coalition-Stable Assignment Is NP-Hard

2026-07-20

AI Summary

This study addresses the existence and computational complexity of coalition-stable matchings in the Hospitals/Residents problem when couples are introduced as inseparable units. By extending classical reduction techniques to the framework of McDermid and Manlove, it establishes for the first time that finding a coalition-stable matching is NP-hard, thereby resolving an open question posed by Rodríguez and Manlove. The hardness persists even under stringent restrictions—specifically, when both hospital capacities and the number of residents are bounded by two—making the result applicable to practical settings such as the HRIC model and course allocation. Additionally, the paper introduces a refined notion of unitwise-coalition stability, offering a new theoretical foundation for future research in this domain.

Abstract

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.

Problem

• coalition stability
• Hospitals/Residents with Inseparable Couples
• NP-hard
• stable assignment
• computational complexity

Innovation

• coalition stability
• NP-hardness
• Hospitals/Residents problem
• inseparable couples
• unitwise-coalition stability