Solution
Unlike the stable marriage problem, the stable roommates may not, in general, have a solution. For a minimal counterexample, consider 4 people A, B, C and D such that the rankings are:
A:(B,C,D), B:(C,A,D), C:(A,B,D), D:(A,B,C)In this ranking, each of A,B,C is the most favorite of someone. In any solution, one of A,B,C must be paired with D and the other 2 with each other (for example AD and BC) yet D's partner and the one for whom D's partner is most favorite would each prefer to be with each other. In this example, AC is a more favorable pairing.
Read more about this topic: Stable Roommates Problem
Famous quotes containing the word solution:
“Who shall forbid a wise skepticism, seeing that there is no practical question on which any thing more than an approximate solution can be had? Is not marriage an open question, when it is alleged, from the beginning of the world, that such as are in the institution wish to get out, and such as are out wish to get in?”
—Ralph Waldo Emerson (1803–1882)
“Coming out, all the way out, is offered more and more as the political solution to our oppression. The argument goes that, if people could see just how many of us there are, some in very important places, the negative stereotype would vanish overnight. ...It is far more realistic to suppose that, if the tenth of the population that is gay became visible tomorrow, the panic of the majority of people would inspire repressive legislation of a sort that would shock even the pessimists among us.”
—Jane Rule (b. 1931)
“I can’t quite define my aversion to asking questions of strangers. From snatches of family battles which I have heard drifting up from railway stations and street corners, I gather that there are a great many men who share my dislike for it, as well as an equal number of women who ... believe it to be the solution to most of this world’s problems.”
—Robert Benchley (1889–1945)