By Jon Kleinberg; Eva Tardos
Advent: a few consultant difficulties -- fundamentals of algorithms research -- Graphs -- grasping algorithms -- Divide and triumph over -- Dynamic programming -- community circulation -- NP and computational intractability -- PSPACE: a category of difficulties past NP -- Extending the boundaries of tractability -- Approximation algorithms -- neighborhood seek -- Randomized algorithms -- Epilogue: algorithms that run endlessly
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Designing the Algorithm We now show that there exists a stable matching for every set of preference lists among the men and women. Moreover, our means of showing this will also answer the second question that we asked above: we will give an efﬁcient algorithm that takes the preference lists and constructs a stable matching. Let us consider some of the basic ideas that motivate the algorithm. Initially, everyone is unmarried. Suppose an unmarried man m chooses the woman w who ranks highest on his preference list and proposes to her.
It would have to implicitly contain algorithms for Interval Scheduling, Bipartite Matching, and a host of other natural optimization problems. The current status of Independent Set is this: no efﬁcient algorithm is known for the problem, and it is conjectured that no such algorithm exists. The obvious brute-force algorithm would try all subsets of the nodes, checking each to see if it is independent, and then recording the largest one encountered. It is possible that this is close to the best we can do on this problem.
W prefers m to m. w prefers m to m . What’s going on in this case? The two men’s preferences mesh perfectly with each other (they rank different women ﬁrst), and the two women’s preferences likewise mesh perfectly with each other. But the men’s preferences clash completely with the women’s preferences. In this second example, there are two different stable matchings. The matching consisting of the pairs (m, w) and (m , w ) is stable, because both men are as happy as possible, so neither would leave their matched partner.