Editing Computational complexity theory (section)

===P versus NP problem===
{{Main|P versus NP problem}}

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the [[Cobham–Edmonds thesis]]. The complexity class [[NP (complexity)|NP]], on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the [[Boolean satisfiability problem]], the [[Hamiltonian path problem]] and the [[vertex cover problem]]. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution.<ref name="Sipser2006">See {{harvnb|Sipser|2006|loc= Chapter 7: Time complexity}}</ref> If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of [[integer programming]] problems in [[operations research]], many problems in [[logistics]], [[protein structure prediction]] in [[biology]],<ref>{{Citation|title=Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete|last1=Berger|first1=Bonnie A.|author1-link=Bonnie Berger|journal=Journal of Computational Biology|year=1998|volume=5|pages=27–40|pmid=9541869|doi=10.1089/cmb.1998.5.27|last2=Leighton|first2=T|author2-link=F. Thomson Leighton|issue=1|postscript=. |citeseerx=10.1.1.139.5547}}</ref> and the ability to find formal proofs of [[pure mathematics]] theorems.<ref>{{Citation|last=Cook|first=Stephen|author-link=Stephen Cook|title=The P versus NP Problem|publisher=[[Clay Mathematics Institute]]|date=April 2000|url=http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf|access-date=2006-10-18|postscript=.|url-status=dead|archive-url=https://web.archive.org/web/20101212035424/http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf|archive-date=December 12, 2010|df=mdy-all}}</ref> The P versus NP problem is one of the [[Millennium Prize Problems]] proposed by the [[Clay Mathematics Institute]]. There is a US$1,000,000 prize for resolving the problem.<ref>{{Citation|title=The Millennium Grand Challenge in Mathematics|last=Jaffe|first=Arthur M.|author-link=Arthur Jaffe|year=2006|journal=Notices of the AMS|volume=53|issue=6|url=https://www.ams.org/notices/200606/fea-jaffe.pdf |archive-url=https://web.archive.org/web/20060612225513/http://www.ams.org/notices/200606/fea-jaffe.pdf |archive-date=2006-06-12 |url-status=live|access-date=2006-10-18|postscript=.}}</ref>