We study a version of the well-known Hospitals/Residents problem in which participants' preferences may involve ties or other forms of indifference. In this context, we investigate the concept of strong stability, arguing that this may be the most appropriate and desirable form of stability in many practical situations. When the indifference is in the form of ties, we describe an O(a 2 ) algorithm to find a strongly stable matching, if one exists, where a is the number of mutually acceptable (resident,hospital) pairs. We also give a lower bound in this case in terms of the complexity of determining whether a bipartite graph contains a perfect matching. By way of contrast, we prove that it becomes NP-complete to determine whether a strongly stable matching exists if the preferences are allowed to be arbitrary partial orders.
In particular, under strong stability, stable matchings might not always exist and it is shown in REF that the problem of deciding whether a strongly stable matching exists, given an instance of the stable marriage problem with partially ordered preferences, is NP-complete.
Strong stability in the hospitals/residents problem
In Proceedings of STACS 2003: the 20th Annual Symposium on Theoretical Aspects of Computer Science
Mathematics Computer Science