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LJ Brain Trust: NP-Complete

stevenredux is working on an article that deals with an NP-complete problem and is having difficulty wrapping his head around it:
I've been trying to understand the concept of an NP-complete problem. I've read about it but I cannot wrap my head around it.

At. All.

Is anyone interested in explaining this to me in English? This is in connection with an article I'm working on, so if you want to take a crack at the explanation please let me know if we're on or off the record.

Thanks.
I know some of my readers probably have enough of an understanding of complexity theory to help him out. If you're so inclined, give him a hand.

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