Lecture 15(b): Illustrative Examples

Here’s an example from the Christiano et al. paper that shows that the width of the basic electrical flow oracle could be as large as {\sqrt{m}}. The effective resistance of the black edges is {1}, so half the current flows on the top red edge. If we set {F = k+1} (which is the max-flow), this means a current of {\Theta(\sqrt{m})} goes on the top edge.

And here’s another example (via Gary, who got this one from Olek Madry) that shows that the fancier algorithm does need {\Omega(m^{1/3})} iterations, and does need to delete {\Omega(m^{1/3})} edges. Again, in this example, {m = \Theta(n)}.

Again, each black gadget has a unit effective resistance, and if you do the calculation, the effective resistance between {s} and {t} tends to the golden ratio. If we set we set {F = n^{1/3}} (which is almost the max-flow), this means a constant fraction of the current, or about {\Theta(n^{1/3})} goes on the edge {e_1}. Once that is deleted, the next red edge {e_2} carries a lot of current, etc. Until all red edges get deleted.

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