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the P vs NP problem attracts a lot of attention, not all of it desirable, for a wide variety of reasons. there are many P=NP claims eg on this widely cited list maintained by mathematician Woegeorgi, P vs NP page. also, intermittently there are hot questions on SE sites related to P vs NP (eg recently [2],[3]) below, & there is even a p-vs-np tag on both cs.se () & tcs.se sites. the following is intended somewhat as a reference question.

what are the basic/typical/common mistakes in P=NP claims?

[1] How not to solve P=NP?, cs.se
[2] P vs NP code exercise, codegolf.se
[3] Analogs of P vs NP in the history of mathematics MO.se


marked as duplicate by D.W., David Richerby, Juho, FrankW, vonbrand Mar 20 '14 at 0:00

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  • $\begingroup$ To me, this seems like a duplicate of [1]. How would this question be different? $\endgroup$ – Juho Mar 19 '14 at 11:10
  • $\begingroup$ agreed similar, feel its not a duplicate because the other question mentions abstract/theoretical principles such as "barriers", BGS relativization, natural proofs etc but doesnt mention more specific errors in the numerous attempts such that authors can check against; the code golf question [2] mentions typical errors in its intro (not mentioned in the supposed dup question). ie other is "top-down" vs this is "bottom-up". $\endgroup$ – vzn Mar 19 '14 at 15:06
  • $\begingroup$ The answers given for [1] are how not to show P!=NP. I thought @vzn was asking specifically about false proofs of P=NP. $\endgroup$ – Austin Buchanan Mar 20 '14 at 1:17

Often, people claim to find a polytime algorithm for an NP-hard problem. A couple things could go wrong:

  1. The algorithm does not return a correct answer.
  2. The algorithm does not run in polytime.

An example of the first mistake is Ted Swart's LP formulations for TSP. They were of poly-size (and so could be solved in polytime with any polytime LP algorithm), but the formulations were not tight. (More info here.)

I have blogged about a particular case of the second mistake. The short story is that the proof of poly runtime "solved" a recurrence relation incorrectly.


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