On P, NP, and computational complexity

TLDR: During and following that exciting week many people have asked me to explain the P vs. NP problem and why it is so important to computer science, and I believe that computational complexity theory sheds limited light on behavior of algorithms in the real world.

Abstract: August 7 and 8, and suddenly the whole world was paying attention. Richard Lipton's August 15 blog entry at blog@ CACM was viewed by about 10,000 readers within a week. Hundreds of computer scientists and mathematicians , in a massive Web-enabled col-laborative effort, dissected the proof in an intense attempt to verify its validity. By the time the New York Times published an article on the topic on August 16, major gaps had been identified, and the excitement was starting to subside. The P vs. NP problem withstood another challenge and remained wide open. During and following that exciting week many people have asked me to explain the problem and why it is so important to computer science. \" If everyone believes that P is different than NP, \" I was asked, \" why it is so important to prove the claim?'' The answer, of course, is that believing is not the same as knowing. The conventional \" wisdom'' can be wrong. While our intuition does tell us that finding solutions ought to be more difficult than checking solutions, which is what the P vs. NP problem is about, intuition can be a poor guide to the truth. Case in point: modern physics. While the P vs. NP quandary is a central problem in computer science, we must remember that a resolution of the problem may have limited practical impact. It is conceivable that P = NP, but the polynomial-time algorithms yielded by a proof of the equality are completely impractical, due to a very large degree of the polynomial or a very large multiplicative constant; after all, (10n) 1000 is a polynomial! Similarly, it is conceivable that P ≠ NP, but NP problems can be solved by algorithms with running time bounded by n log log log n —a bound that is not polynomial but incredibly well behaved. Even more significant, I believe, is the fact that computational complexity theory sheds limited light on behavior of algorithms in the real world. Take, for example, the Boolean Satisfi-ability Problem (SAT), which is the ca-nonical NP-complete problem. When I was a graduate student, SAT was a \" scary \" problem, not to be touched with a 10-foot pole. Garey and John-son's classical textbook showed a long sad line of programmers who have failed to solve NP-complete problems. Guess what? These programmers have been busy! The August 2009 issue of Communications contained …