Stable roommates problem

The stable roommates problem asks how to pair an even number of people, each with a ranked list of preferences, so no two prefer each other over their assigned partners.

Preference list: An ordered ranking of acceptable choices for a participant in a matching problem. Each person’s ranking determines which pairs can block a proposed roommate assignment.
Stable marriage problem: A matching problem with two separate groups, where every participant ranks members of the other group. Unlike the roommate problem, its two-sided structure guarantees a stable matching for complete preferences.
Three-person stable roommates problem: A roommate instance with three participants, which cannot pair everyone without leaving one unmatched. It shows why the even-number requirement is necessary for a complete pairing.
Roommate assignment: The allocation of people to shared living arrangements, often using compatibility or preference information. It is the most direct setting represented by the problem’s name and pairing structure.
Computational complexity: The study of the resources required to solve computational problems as their inputs grow. It frames why polynomial-time solvability matters for large roommate instances.
Blocking pair: A pair of participants who both prefer each other to their assigned partners. A stable roommate assignment exists precisely when it has no blocking pair.
Gale–Shapley algorithm: An algorithm that finds a stable matching between two groups through repeated proposals and rejections. It solves the two-sided marriage problem, but its method does not directly solve general roommate instances.
Four-person stable roommates problem: A roommate instance with four participants and rankings over potential partners. A small profile can demonstrate that even-sized instances may still have no stable pairing.
Stable fixtures problem: A matching problem in which participants may form groups of three under ranked preferences. It generalizes roommate pairing from pairs to three-person groups.
NP-completeness: A classification for decision problems that are among the hardest problems in NP under efficient reductions. Many variants of stable matching become computationally hard even when the classical problem is tractable.

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