NP-complete problems are studied because the ability to quickly verify solutions to a problem (NP) seems to correlate with the ability to quickly solve that problem P (complexity). It is not known whether every problem in NP can be quickly solved—this is called the NP-complete. But if any single problem in NP-complete can be solved quickly, then every problem in NP can also be quickly solved, because the definition of an NP-complete problem states that every problem in NP must be quickly reducible to every problem in NP-complete (that is, it can be reduced in polynomial time). Because of this, it is often said that the NP-complete problems are harder or more difficult than NP problems in general. Formal definition of NP-completeness Main article: formal definition for NP- completeness. A decision problem upload.wikimedia.org/math/f/c/7/ fc783207375c0abb99cfb79841b1708d.png is NP-complete if: upload.wikimedia.org/math/f/c/7/fc783207375c0abb99cfb79841b1708d.png is in NP, and Every problem in NP is a Reduction (complexity) upload.wikimedia.org/math/f/c/7/ fc783207375c0abb99cfb79841b1708d.png in polynomial time. upload.wikimedia.org/math/f/c/7/fc783207375c0abb99cfb79841b1708d.png can be shown to be in NP by demonstrating that a candidate solution to upload.wikimedia.org/math/f/c/7/fc783207375c0abb99cfb79841b1708d.png can be verified in polynomial time. A problem pload.wikimedia.org/math/5/8/a/58a0d43f50180ea2c46c506e745f0e8d.png is reducible to upload.wikimedia.org/math/f/c/7/fc783207375c0abb99cfb79841b1708d.png if there is a polynomial-time many-one reduction, a Deterministic algorithm which transforms any instance upload.wikimedia.org/math/f/f/3/ ff350c15776a952c12dde81ac71c5d15.png upload.wikimedia.org/math/c/c/d/ccd054fd2c7efcd5d40f843387eaf1c2.png, such that the answer to upload.wikimedia.org/math/ 7/4/8/748313066baf5e80e875d513a18604cd.png if and only if the answer to upload.wikimedia.org/math/6/d/ 2/6d2638f3b017bed72452bddbc28cbd6a.png To prove that an NP problem upload.wikimedia.org/math/f/c/7/fc783207375c0abb99cfb79841b1708d.png is in fact an NP-complete problem it is sufficient to show that an already known NP-complete problem reduces to upload.wikimedia.org/math/f/c/7/ fc783207375c0abb99cfb79841b1708d.png Note that a problem satisfying condition 2 is said to be whether or not it satisfies condition 1. A consequence of this definition is that if we had a polynomial time algorithm (on a Universal Turing machine UTM or any other Turing-equivalent abstract machine for upload.wikimedia.org/math/f/c/7/fc783207375c0abb99cfb79841b1708d.png, we could solve all problems in NP in polynomial time. =NP- complete&action=edit& Edit section: Background span class mw-headline The concept of NP-complete was introduced in 1971 by Stephen Cook in a paper entitled The Complexity of theorem-proving procedures. on pages 151-158 of the Proceedings of the 3rd Annual ACM Symposium on Theory of Computing though the term did not appear anywhere in his paper. At that computer science conference, there was a fierce debate among the computer scientists about whether NP-complete problems could be solved in polynomial time on a Deterministic class mw-Turing machine. John Hopcroft brought everyone at the conference to a consensus that the question of whether NP-complete problems are Run time (computing) solvable in polynomial time should be put off to be solved at some later date, since nobody had any formal proofs for their claims one way or the other. This is known as the question of whether P=NP. Nobody has yet been able to determine conclusively whether NP- complete problems are in fact solvable in polynomial time, making this one of the great unsolved problems of mathematics. Clay Mathematics Institute is offering a US$1 million reward to anyone who has a formal proof that P=NP or that P≠NP. In the celebrated Cooks theorem class (independently proved by Leonid Cook proved that the Boolean satisfiability problem is NP-complete (a simpler, but still highly technical Proof that Boolean satisfiability problem is NP-complete proof of this is available). In 1972, Richard Karp proved that several other problems were also NP-complete. Karps 21 NP-complete problems thus there is a class of NP-complete problems (besides the Boolean satisfiability problem). Since Cooks original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in a Michael Garey and David S. Johnsons 1979 book Computers and Intractability: A Guide to NP-Completeness
Posted on: Mon, 24 Nov 2014 04:29:06 +0000
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