Open Problems in the area of Matching Under Preferences

Matching problems span several academic disciplines including Computer Science, Mathematics and Economics. In 2012, the Nobel Prize for Economic Sciences was awarded to Lloyd S. Shapley and Alvin E. Roth for their work in this area.
Matching problems arise when it is necessary to assign one or more sets of entities to each other. Practical applications include the assignment of kidney donors to patients; graduating medical students to hospitals or applicants to campus housing accommodation.
In the case of assigning a set of graduating medical students to a set of hospitals, it is clear that both sets may have preferences over the other. Medical students may have preference over hospitals depending on hospital reputation or recommendation from other students and hospitals may have preference over students due to grades achieved in their studies. A matching in this case may be defined as an assignment of medical students to hospitals such that a medical student is assigned at most once, and hospitals are only assigned as many students as they can hold. Problem instances, such as this, may admit many different matchings and which matching is "best" or optimal depends on the priorities of the matching scheme administrator. It may be a priority to match as many medical students to hospitals as possible, or to maximise the number of medical students who get their first choice hospital, and subject to this, their second choice hospital, and so on.
Over several decades, efficient algorithms have been developed to find different types of optimal matchings. As discussed previously, many of these algorithms are used in real world situations, such as the National Resident Matching Program (NRMP) [2], a matching scheme in the US which matches graduating medical students to hospitals and the National Living Donor Kidney Sharing Scheme (NLDKSS) provided by the NHS which uses an algorithm developed by the University of Glasgow [1] to match kidney donors to kidney patients.
My proposed PhD work focusses on generating new structural and algorithmic results for matching problems with different optimality criteria. A rank-maximal matching is one which maximises the number assigned to their first choice and, subject to this, the number assigned to their second choice, and so on. I will seek new efficient algorithms to enumerate all rank-maximal matchings and find pairs that belong to rank-maximal matchings. Similar opportunities are available for greedy maximum matchings in which the maximum number of people are matched as possible, and subject to this, we optimise on first, second, and third choices, as in rank-maximal matchings. A further optimality criterion, popularity, is defined for matchings in which no other matching is preferred by a majority who have a strict preference between them. New algorithms will be sought to count and enumerate popular matchings with additional optimality criteria and to find pairs belonging to those matchings. By using efficient algorithms to count and enumerate optimal matchings, and to find pairs belonging to them, matching scheme administrators will be given more flexibility in determining a "best possible" matching, in their matching scenario.

References
[1] David F. Manlove and Gregg O'Malley. Paired and Altruistic Kidney Donation in the UK: Algorithms and Experimentation. Experimental Algorithms, 7276:271-282, 2012.
[2] Elliott Peranson and Richard R. Randlett. The NRMP matching algorithm revisited: Theory versus practice. Academic Medicine, 70(6):477-484, 1995.