Newsflash: complexity remains complex

August 10, 2010 § Leave a comment

For the computer scientists among you: you may be interested to know that a claim of a proof that P is not equal to NP has been getting a lot of attention on compsci blogs.

For the non computer scientists (and non-mathematicians), Wikipedia’s translation of the P=NP problem is as follows.  Suppose that “yes”-answers to a “yes”-or-“no”-question can be verified “quickly”. Then can the answers themselves also be computed “quickly”? [Quickly, of course, doesn’t quite mean what you or I would think it does, but anyway, you get the idea.]

According to this new proof, P≠NP and so the answer is no.  That’s not a huge surprise — most people weren’t hopeful that the answer would go the other way. (If it turned out that P did equal NP, then one could imagine using the insights from the proof to help solve a wide range of currently intractable problems.)  But since P=NP is one of the seven great unsolved problems in mathematics chosen by the Clay Mathematics Institute, it would be big news if the proof were to stand.  And a real proof of P≠NP would surely have many important consequences.

The Clay Institute is offering $1M prizes for each of the confirmed proofs, and has already awarded one of the seven prizes to the reclusive Russian mathematician Grigori Perelman for his proof of the Poincaré conjecture.  They notified him of his prize in March this year: he turned them down.  A few years ago, he turned down the Fields Medal too.  Nice to see the old stereotype of the antisocial mathematician being so successfully defended.

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