By Jon Kleinberg; Eva Tardos
Advent: a few consultant difficulties -- fundamentals of algorithms research -- Graphs -- grasping algorithms -- Divide and overcome -- Dynamic programming -- community circulation -- NP and computational intractability -- PSPACE: a category of difficulties past NP -- Extending the bounds of tractability -- Approximation algorithms -- neighborhood seek -- Randomized algorithms -- Epilogue: algorithms that run without end
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And for each, we can ask about the existence of stable matchings, as follows. (a) A strong instability in a perfect matching S consists of a man m and a woman w, such that each of m and w prefers the other to their partner in S. Does there always exist a perfect matching with no Exercises strong instability? Either give an example of a set of men and women with preference lists for which every perfect matching has a strong instability; or give an algorithm that is guaranteed to find a perfect matching with no strong instability.
And in larger examples, with more than two people on each side, we can have an even larger collection of possible stable matchings, many of them not achievable by any natural algorithm. This example shows a certain “unfairness” in the G-S algorithm, favoring men. If the men’s preferences mesh perfectly (they all list different women as their ﬁrst choice), then in all runs of the G-S algorithm all men end up matched with their ﬁrst choice, independent of the preferences of the women. If the women’s preferences clash completely with the men’s preferences (as was the case in this example), then the resulting stable matching is as bad as possible for the women.
Otherwise, w is already engaged to some other man m . In this case, she determines which of m or m ranks higher on her preference list; this man becomes engaged to w and the other becomes free. Finally, the algorithm will terminate when no one is free; at this moment, all engagements are declared ﬁnal, and the resulting perfect matching is returned. 2 depicting a state of the algorithm. Initially all m ∈ M and w ∈ W are free While there is a man m who is free and hasn’t proposed to every woman Choose such a man m Let w be the highest-ranked woman in m’s preference list to whom m has not yet proposed If w is free then (m, w) become engaged Else w is currently engaged to m If w prefers m to m then m remains free Else w prefers m to m (m, w) become engaged m becomes free Endif Endif Endwhile Return the set S of engaged pairs An intriguing thing is that, although the G-S algorithm is quite simple to state, it is not immediately obvious that it returns a stable matching, or even a perfect matching.