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 . The effective resistance of the black edges is , so half the current flows on the top red edge. If we set (which is the max-flow), this means a current of 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 iterations, and does need to delete edges. Again, in this example, .

Again, each black gadget has a unit effective resistance, and if you do the calculation, the effective resistance between and tends to the golden ratio. If we set we set (which is almost the max-flow), this means a constant fraction of the current, or about goes on the edge . Once that is deleted, the next red edge carries a lot of current, etc. Until all red edges get deleted.

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