P versus NP is a contender for the most important unsolved problem in math. For those tuning in from home, P stands for Polynomial Time. It’s the class of all yes-or-no problems that a digital computer can solve “efficiently”—meaning, using a number of steps that grows at most like the number of bits needed to specify the problem raised to some fixed power. Some examples are: I give you a map, and I ask whether every town is at most 200 miles from every other. Or I give you a positive integer, and I ask whether it’s prime. NP stands for Nondeterministic Polynomial-Time. It’s the class of yes-or-no problems for which, if the answer is “yes,” there’s a short proof that a computer can efficiently check. An example of an NP problem is: I give you a positive integer, and I ask whether it has at least five divisors. No one knows a fast algorithm for the latter problem: indeed, the presumed hardness of this sort of problem (for classical computers, anyway!) is the basis for most modern cryptography. Still, if the answer is “yes,” you could prove it to someone by just showing them the divisors.

scottaaronson
physics
math
p=np
thesingularity
april 2016 by trevf

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