For this, suppose we have a network of cities connected by roads. If you are given a filled-in Sudoku puzzle in a newspaper, you might chuck it in the bin. (Imagine a jigsaw puzzle: finding the right arrangement of pieces is difficult, but you can tell when the puzzle is finished correctly just by looking at it.). Addendum. cartoon "The History and Status of the P versus NP question", A Short History of Computational Complexity, NP-complete Problems and Physical Reality, P, NP and Mathematics - a computational complexity perspective, http://www.claymath.org/millennium-problems, http://www.business.uconn.edu/users/mdiaby/tsplp/, http://www.business.uconn.edu/users/mdiaby/qaplp/, Linear programming formulation of the vertex colouring problem, Linear programming formulation of the set partitioning problem, http://www.worldscientific.com/worldscibooks/10.1142/9725, http://www.tarusa.ru/~mit/ENG/sigma01_e.php, http://www.wbabin.net/science/mimouni.pdf, http://www.scialert.net/pdfs/itj/2006/851-859.pdf, http://www.freepatentsonline.com/y2008/0071849.html, http://www.waset.org/pwaset/v35/v35-74.pdf, "Refutation of Aslam's Proof that NP = P", "Response to Refutation of Aslam's Proof that NP = P", http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/pnp.pdf, http://trytoprovenpvsp.blog.sohu.com/entry/, http://www.isrn.com/journals/cm/2012/321372/, http://sites.google.com/site/ariblindercswork/, http://romvf.wordpress.com/2011/01/19/open-letter/, http://www.iigss.net/Scientific-Inquiry/Dec2010/table.htm, http://the-point-of-view-of-frank.blogspot.com/, http://link.springer.com/chapter/10.1007/978-94-007-7684-5_22/, https://hal.archives-ouvertes.fr/hal-01161668, https://hal.archives-ouvertes.fr/hal-01270398. 1.1 The Importance of P =? reasons The input is a given length, say n, where n is a positive integer. Finding a solution to a problem should be harder than checking that a solution is correct. The problem is to visit each city using roads (no planes allowed!) If you know of other papers in this area, then please send me the links. parody suggested cannot work. For example, it is unknown whether NL is a proper subclass of P, and we don’t know if FPT is a proper subclass of W[P]. An NP–complete problem is one which is NP-hard and in NP. P =? But as a species, dumb is our default position so no one can say what the answer is with 100% certainty. Given the recent interest in the problem and its central place in mathematics, I discuss it in this week’s blog. (picture P and NP are two groups of mathematical problems. More precisely, the P-versus-NP problem is shown to be a scientific rather than a mathematical problem. “Even though we’ve been writing algorithms for decades, we don’t fully understand what they’re capable of,” he continues. where finding the answer is easy. A generic instance of TSP consists of a finite set This seems easy for a small number of cities, but if you were given hundreds of them then it becomes hard.