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 ceaselessly
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For a while, no one has proposed to her, and she is free. Then a man m may propose to her, and she becomes engaged. As time goes on, she may receive additional proposals, accepting those that increase the rank of her partner. So we discover the following. 1) w remains engaged from the point at which she receives her ﬁrst proposal; and the sequence of partners to which she is engaged gets better and better (in terms of her preference list). The view of a man m during the execution of the algorithm is rather different.
Finally, suppose there is an instability of type (iv), consisting of a man m and a woman w, neither of which is part of any pair in the matching, so that (m, w) ∈ F. But for m to be single, he must have proposed to every nonforbidden woman; in particular, he must have proposed to w, which means she would no longer be single—a contradiction. Exercises 1. Decide whether you think the following statement is true or false. If it is true, give a short explanation. If it is false, give a counterexample.
Here’s a natural question: Is there anything good we can say about the complexity of the Independent Set Problem? One positive thing is the following: If we have a graph G on 1,000 nodes, and we want to convince you that it contains an independent set S of size 100, then it’s quite easy. We simply show you the graph G, circle the nodes of S in red, and let you check that no two of them are joined by an edge. So there really seems to be a great difference in difﬁculty between checking that something is a large independent set and actually ﬁnding a large independent set.