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Topic: NP (complexity)

	Complexity classes P and NP - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem.
		Furthermore, the result P = NP would imply many other startling results that are currently believed to be false, such as NP = co-NP and P = PH.
		Similarly, NP is the set of languages expressible in existential second-order logic — that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets.
	en.wikipedia.org /wiki/Complexity_classes_P_and_NP (2854 words)

	NP (complexity) - Wikipedia, the free encyclopedia
		In computational complexity theory, NP ("Non-deterministic Polynomial time") is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine.
		NP can be seen as a very simple type of interactive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic polynomial-time machine that checks it.
		A major result of complexity theory is that NP can be characterized as the problems solvable by probabilistically checkable proofs where the verifier uses O(log n) random bits and examines only a constant number of bits of the proof string (the class PCP(log n, 1)).
	en.wikipedia.org /wiki/NP_(complexity) (1073 words)

	NP-complete
		In complexity theory, the complexity class NP-complete is the set of problems that are the hardest problems in NP, in the sense that they are the ones most likely not to be in P.
		Assuming that P and NP are not equal, there are guaranteed to be an infinite number of problems that are in NP, but are neither NP-complete nor in P. Some of these problems may actually have higher complexity than some of the NP-complete problems.
		This implies that NP = co-NP as is shown in the proof in the article on co-NP.
	www.ebroadcast.com.au /lookup/encyclopedia/np/NP-complete_problems.html (969 words)

	Computational complexity theory From Wikipedia (Site not responding. Last check: 2007-11-07)
		Complexity Theory is a part of the theory of computation dealing with the resources required during computation to solve a given problem.
		Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required.
		The time complexity of a problem is the number of steps that it takes to solve an instance, as a function of the size of the instance.
	www.xtrj.org /ssm4/complexity_theory.htm (765 words)

	Complexity: N and NP
		In the field of complexity theory, there is a monolothic question that has baffled computer scientists for some time now.
		When dealing with problems and their complexity, often, it is better to view the problem as a language and use what we call a Turing machine to help us gauge the problem's complexity.
		NP is the set of all languages, L, such that there exists a non-deterministic Turing machine, M, such that M can accept every instance of L in polynomial time.
	cgm.cs.mcgill.ca /~cwu25/proj507/pandnp.html (667 words)

	NOTES ON COMPLEXITY AND NP COMPLETENESS (Site not responding. Last check: 2007-11-07)
		Another is that we know of a number of problems (such as SAT) that are in NP (ie, that we can verify in polynomial time) and that we do not know how to decide in polynomial time.
		all other problems q in NP can be "reduced in polynomial time" to solving p: there is a function f computable in polynomial-time such that for all inputs i to p, the correct answer to q(i) is yes iff the correct answer to p(f(i)) is yes.
		And of course, P is a subset of NP.
	www.cs.umd.edu /class/spring2003/cmsc351/notes/complexity.html (1866 words)

	Dieter van Melkebeek - Research on Computational Complexity Theory
		Separating P from NP is equivalent to showing that satisfiability -- the problem of deciding whether there is a setting of the variables in a given propositional formula that makes it true -- cannot be solved in polynomial time.
		Torenvliet established that large complexity classes like doubly exponential space have complete languages that are not autoreducible, whereas the complete languages of smaller classes like exponential time all share the property of autoreducibility.
		The Kolmogorov complexity of a string is the length of its shortest description; various complexity restrictions on the descriptions lead to various notions of Kolmogorov complexity.
	www.cs.wisc.edu /~dieter/Research/complexity.html (2933 words)

	Complexity Theory
		The time complexity of an algorithm is one of the most common determinants of the success of a computational strategy, although occasionally one finds that exorbitant amounts of energy or incredible numbers of particles can be substituted for a long amount of time.
		Determining the complexity class of a program is difficult because of the negative nature of this definition: one must show that no one can make a faster program.
		There is a class of problems known as NP problems, which can only be solved by exponential time (brute-force) style algorithms, but nobody has proven that they cannot be solved in polynomial time.
	www.media.mit.edu /physics/pedagogy/babbage/texts/ct.html (2297 words)

	Complexity Zoo - Qwiki
		In descriptive complexity, uniform AC can be characterized as the class of problems expressible by first-order predicates with addition and multiplication operators - or indeed, with ordering and multiplication, or ordering and division (see [Lee02]).
		NP, and indeed NP intersect coNP, are not contained in BQP [BBB97].
		The class of decision problems L in NP such that, if the answer is "yes," then a proof can be constructed in polynomial time given access only to an oracle for L. Contains NPC.
	qwiki.caltech.edu /wiki/Complexity_Zoo (6334 words)

	Np complexity - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		Look for Np complexity in Wiktionary, our sister dictionary project.
		Look for Np complexity in the Commons, our repository for free images, music, sound, and video.
		Check for Np complexity in the deletion log, or visit its deletion vote page if it exists.
	www.sciencedaily.com /encyclopedia/np__complexity_ (145 words)

	Wikinfo \| Computational complexity theory
		The time complexity of a problem is the number of steps that it takes to solve an instance of the problem, as a function of the size of the input, (usually measured in bits) using the most efficient algorithm.
		The set NP is the set of decision problems that can be solved by a non-deterministic machine in polynomial time.
		Many of these classes have a 'Co' partner (ie NP and Co-NP) which consists of the complements of all languages in the original class.
	www.wikinfo.org /wiki.php?title=Computational_complexity_theory (1042 words)

	Complexity Zoo References (Site not responding. Last check: 2007-11-07)
		The complexity of theorem-proving procedures, Proceedings of ACM STOC'71, pp.
		On the parameteric complexity of relational database queries and a sharper characterization of W[1], Combinatorics, Complexity, and Logic, Proceedings of DMTCS'96, Springer-Verlag, pp.
		Hilbert's Nullstellensatz is in the polynomial hierarchy, Journal of Complexity 12(4):273-286, 1996, DIMACS TR 96-27.
	www.complexityzoo.com /zooref.html (5760 words)

	NP-Completeness
		A problem is in NP if you can quickly (in polynomial time) test whether a solution is correct (without worrying about how hard it might be to find the solution).
		Problems in NP are still relatively easy: if only we could guess the right solution, we could then quickly test it.
		So if we believe that P and NP are unequal, and we prove that some problem is NP-complete, we should believe that it doesn't have a fast algorithm.
	www.ics.uci.edu /~eppstein/161/960312.html (3273 words)

	Complexity classes P and NP (Site not responding. Last check: 2007-11-07)
		Complexity classes P and NP Complexity classes P and NP
		If '''P''' = '''NP''', then all three classes are equal.]] [[Computational complexity theory]] is part of the [[theory of computation]] dealing with the resources required during computation to solve a given problem.
		While this is a common and reasonably accurate assumption in complexity theory, it is not always true in practice, for several reasons:*It ignores constant factors.
	complexityclassespandnp.quickseek.com (2038 words)

	Stas Busygin's NP-Completeness Page (Site not responding. Last check: 2007-11-07)
		In 60's-70's of the last century there was developed a conception designating a hierarchy of complexity classes for problems on finite sets.
		With the current state-of-the-art the most important complexity classes are P (problems solvable in polynomial time) and NP (problems whose solution certificate can be verified in polynomial time).
		And it is also in NP as when someone gives us another list asserting it is a sorted version of the first one, we can easily look through it to check the order and compare elements to verify are they the same as initial.
	www.busygin.dp.ua /npc.html (896 words)

	Complexity classes P and NP Info - Encyclopedia WikiWhat.com (Site not responding. Last check: 2007-11-07)
		This means that if a single NP-complete problem could be shown to be in P, then it would follow that P = NP.
		This means it requires exponential time, and so is outside P and NP.
		The problem of deciding the truth of a statement in Presburger arithmetic is even harder.
	www.wikiwhat.com /encyclopedia/c/co/complexity_classes_p_and_np.html (1529 words)

	Proof Complexity
		NP consists of those languages that have short, easily-verifiable proofs of membership.
		Proof complexity is the study of the lengths of easily-verifiable proofs for co-NP languages.
		The classifications given by proof complexity and lower bounds on the sizes of these proofs permit us to show that large classes of deterministic algorithms require exponential time to solve the problems.
	www.cs.washington.edu /homes/beame/projects/proofcomplexity.html (756 words)

	Average-Case Complexity Forum (Site not responding. Last check: 2007-11-07)
		Indeed, although NP-complete problems are generally thought of as being computationally intractable, some are easy on average; and some are complete in the average case, indicating that they remain difficult on randomly generated instances.
		This forum provides an overview of the recent research on average complexity, and shows the subtleties in formulating a coherent framework for studying average-case NP-completeness.
		Average-Case Intractible NP Problems, a survey of average-case NP-complete problems.
	www.uncg.edu /mat/avg.html (233 words)

	Co-NP (Site not responding. Last check: 2007-11-07)
		Since all problems in '''NP''' can be reduced to this problem it follows that for all problems in '''NP''' we can construct a non-deterministic Turing machine that decides the complement of the problem in polynomial time, i.e., '''NP''' is a subset of '''co-NP'''.
		From this it follows that the set of complements of the problems in '''NP''' is a subset of the set of complements of the problems in '''co-NP''', i.e., '''co-NP''' is a subset of '''NP'''.
		It is in both '''NP''' and '''co-NP''', but is generally suspected to be outside '''P''', outside '''NP'''-complete, and outside '''co-NP'''-complete.
	conp.quickseek.com (316 words)

	Complexity Theory - NP and NP-completeness
		NP = { f : CHECK(f) is in P }
		NP is the class of problems with efficient `verifying' algorithms.
		With the exception of the last 2, it is generally believed that no efficient algorithm exists for any of these decision problems.
	www.csc.liv.ac.uk /~ped/teachadmin/algor/npcomp.html (1801 words)

	Complexity Theory
		The aim of the course is to introduce the theory of computational complexity.
		The course will explain measures of the complexity of problems and of algorithms, based on time and space used on abstract models.
		Important complexity classes will be defined, and the notion of completeness established through a thorough study of NP-completeness.
	www.cl.cam.ac.uk /DeptInfo/CST03/node109.html (143 words)

	Learn more about NP (complexity) in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		Learn more about NP (complexity) in the online encyclopedia.
		Or, equivalently, YES answers are checkable in polynomial time on a deterministic Turing machine given the right information.
		See also: Complexity classes P and NP and NP-Complete.
	www.onlineencyclopedia.org /n/np/np__complexity_.html (170 words)

	URCS Theory Technical Reports
		While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility---so-called selector functions---can without cost be chosen to possess such algebraic properties as commutativity and associativity.
		We study the complexity of quantum complexity classes such as EQP, BQP, and NQP (quantum analogs of P, BPP, and NP, respectively) using classical complexity classes such as ZPP, WPP, and C_{=}P. The contributions of this paper are threefold.
		Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [Ogihara and Hemachandra 1993], we nonetheless show that it often is possible to reduce the number of solutions of NP functions.
	www.cs.rochester.edu /trs/theory-trs.html (16214 words)

	COT 5410. Complexity of Algorithms
		We show that NP complete problems are in some sense the hardest problems that we can at least hope to solve efficiently, and give reasons why it is commonly believed that they cannot be solved efficiently.
		NP complete problems arise in many real-life applications, such as compilers, scheduling, and parallel computation.
		Given an application, abstract its computational structure and either prove that it is NP hard, or give a polynomial time algorithm to solve it.
	www.cs.fsu.edu /~asriniva/courses/alg04 (2627 words)

	Computability and Complexity
		The Space Complexity of a decision problem, f, is the amount of memory used by the `best' algorithm A for f.
		A major open problem in Computational Complexity Theory is to develop arguments by which important computational problems can have their time complexity described exactly.
		Computational Complexity Theory involves a large number of subfields each of which is ultimately concerned with problems such as those above, e.g.
	www.csc.liv.ac.uk /~ped/teachadmin/algor/comput_complete.html (5669 words)

	Complexity Theory Lecture Notes (summaries)
		Loosely speaking, the first condition (i.e., efficient verification) is captured in the definition of NP, and the second in that of P. The actual correspondence relies on the notion of self-reducibility, which relates the complexity of determining whether a solution exists to the complexity of actually finding one.
		We define space complexity using an adequate model of computation in which one is not allowed to use the area occupied by the input for computation.
		We then relate this hierarchy to complexity classes discussed in previous lectures such as BPP and P/poly: We show that BPP is in PH, and that if NP subseteq P/poly then PH collapses to is second level.
	www.eccc.uni-trier.de /eccc-local/ECCC-LectureNotes/IntroComplTh/cc-sum.html (2064 words)

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