NP-Complete Stable Matching Problems

This paper concerns the complexity analysis of the roommate problem and intern assignment problem with couples. These are two special cases of the matching problems known as stable matching. The roommate problem is that of assigning a set of people to rooms of exactly two occupants in accordance with the preferences of the members of the set. The intern assignment problem with couples is that of assigning a set of graduating medical students as interns in hospital residency programs, in accordance to their priorities and the priorities of the programs. Some of the interns may be married and in this case every such couple has preferences over pairs of joint assignments. These assignments must satisfy certain stability requirements. The problems can be defined so that, when stating their preferences, these participating agents can give the same (tied) preference to agents with whom they can be matched. This paper shows that the option of allowing ties in the preference lists can significantly affect the computational complexity of stable matching. It will be shown that, when ties are allowed, the roommate problem is NP-complete. In contrast, the roommate problem without ties is known to be solvable in polynomial time. In addition, this paper will demonstrate the NP-completeness of the intern assignment problem with couples (even without ties).