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A fact from Widest path problem appeared on Wikipedia's Main Page in the Did you know column on 3 May 2011 (check views). The text of the entry was as follows:
As far as I know, the "width" of an arc is usually called its "capacity". I recommend that this article should be moved to "maximum capacity path problem". Markus Schulze16:27, 27 April 2011 (UTC)[reply]
My guess prior to looking at the actual numbers was that "bottleneck shortest path" would be a more frequently used name than the one you propose. But Google scholar finds 1000 hits for "widest path", compared to 86 for "bottleneck shortest path" and 195 for "maximum capacity path", so (if the hit count can be believed) the current name is actually a good choice. I did add your alternative to the lede, though. —David Eppstein (talk) 17:13, 27 April 2011 (UTC)[reply]