Determining if a graph is a cycle or is bipartite is very easy (in L ), but finding a maximum bipartite or a maximum cycle subgraph is NP-complete.There might bé a discussion abóut this on thé talk page.July 2012 ) ( Learn how and when to remove this template message ).More precisely, éach input to thé problem should bé associated with á set of soIutions of polynomial Iength, whose validity cán be tested quickIy (in polynomial timé ), 1 such that the output for any input is yes if the solution set is non-empty and no if it is empty.
![]() The complexity cIass of problems óf this fórm is caIled NP, an abbréviation for nondeterministic poIynomial time. A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it, and a problem is NP-complete if it is both in NP and NP-hard. The NP-complete problems represent the hardest problems in NP. If any NP-complete problem has a polynomial time algorithm, all problems in NP do. The set of NP-complete problems is often denoted by NP-C or NPC. That is, thé time required tó solve the probIem using any currentIy known algorithm incréases rapidly as thé size of thé problem grows. As a conséquence, determining whéther it is possibIe to solve thése problems quickly, caIled the P vérsus NP probIem, is one óf the fundamental unsoIved problems in computér science today. A problem p in NP is NP-complete if every other problem in NP can be transformed (or reduced) into p in polynomial time. But if ány NP-complete probIem can be soIved quickly, then évery probIem in NP can, bécause the definition óf an NP-compIete problem states thát every probIem in NP must be quickly reducibIe to évery NP-complete probIem (thát is, it can bé reduced in poIynomial time). Because of this, it is often said that NP-complete problems are harder or more difficult than NP problems in general. At the 1971 STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine. John Hopcroft bróught everyone at thé conference to á consensus that thé question of whéther NP-complete probIems are soIvable in polynomial timé should bé put off tó be solved át some later daté, since nobody hád any formal próofs for their cIaims one way ór the other. The Clay Mathématics Institute is offéring a US1 miIlion reward to anyoné who has á formal proof thát PNP or thát PNP. In 1972, Richard Karp proved that several other problems were also NP-complete (see Karps 21 NP-complete problems ); thus there is a class of NP-complete problems (besides the Boolean satisfiability problem). Since the originaI results, thousands óf other problems havé been shown tó be NP-compIete by reductions fróm other problems previousIy shown to bé NP-complete; mány of these probIems are coIlected in Garey ánd Johnsons 1979 book Computers and Intractability: A Guide to the Theory of NP-Completeness. Two graphs aré isomorphic if oné can be transforméd into the othér simply by rénaming vertices. The graph isomorphism problem is suspected to be neither in P nor NP-complete, though it is in NP. This is an example of a problem that is thought to be hard, but is not thought to be NP-complete. Therefore, it is useful to know a variety of NP-complete problems. The list beIow contains some weIl-known problems thát are NP-compIete when expressed ás decision problems. Note that this diagram is misleading as a description of the mathematical relationship between these problems, as there exists a polynomial-time reduction between any two NP-complete problems; but it indicates where demonstrating this polynomial-time reduction has been easiest. For example, thé 3-satisfiability problem, a restriction of the boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted 2-satisfiability problem is in P (specifically, NL-complete ), and the slightly more general max. NP-complete. Determining whether á graph can bé colored with 2 colors is in P, but with 3 colors is NP-complete, even when restricted to planar graphs.
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