By John Mount
Uwe Kils “Iceberg”
In this article I am attempting to reproduce some fraction of the insight found in:
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).
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).
Introduction
Reading a mathematics paper is a very active task. Proofs are often abridged (requiring the reader to re-invent sections to follow along), long (requiring a reader to check things sequentially), or applications of diffuse networks of lemmas (requiring the reader to run all over the paper and …read more
Source:: win-vector.com