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Reading and writing proofs

In my recent article on optimizing set diversity I mentioned the primary abstraction was of “diminishing returns” and is formalized by the theory of monotone submodular functions (though I did call out some of my own work which used a different abstraction). A proof that appears again and again in the literature is: showing that when maximizing a monotone submodular function the greedy algorithm run for k steps picks a set that is scores no worse than 1-1/e less than the unknown optimal pick (or picks up at least 63% of the possible value). This is significant, because naive optimization may only pick a set of value 1/k of the value of the optimal selection.

The proof that the greedy algorithm does well in maximizing monotone increasing submodular functions is clever and a very good opportunity to teach about reading and writing mathematical proofs. The point is: one needs an active reading style as: most of what is crucial to a proof isn’t written, and that which is written in a proof can’t all be pivotal (else proofs would be a lot more fragile than they actually are).

Uwe Kils “Iceberg”

In this article I am attempting to reproduce some fraction of the insight found in: Polya “How to Solve It” (1945) and Doron Zeilberger “The Method of Undetermined Generalization and Specialization Illustrated with Fred Galvin’s Amazing Proof of the Dinitz Conjecture” (1994).

So I repeat the proof here (with some annotations and commentary). Continue reading Reading and writing proofs