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	Kids.Net.Au - Encyclopedia > NP-complete (Site not responding. Last check: 2007-10-21)
		One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any nonempty subset of them adds up to zero.
		The Graph Isomorphism problem is suspected to be neither in P nor NP-complete, though it is obviously in NP.
		A problem X is polynomial time, Turing reducible to a problem Y if, given a subroutine that solves Y in polynomial time, you could write a program that calls this subroutine and solves X in polynomial time.
	www.kids.net.au /encyclopedia-wiki/np/NP-complete (974 words)

	NP-complete - Wikipedia, the free encyclopedia
		One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them sums to zero.
		An interesting example is the graph isomorphism problem, the graph theory problem of determining whether a graph isomorphism exists between two graphs.
		The Graph Isomorphism problem is suspected to be neither in P nor NP-complete, though it is obviously in NP.
	en.wikipedia.org /wiki/NP-complete (1450 words)

	NationMaster - Encyclopedia: Independent Set problem
		The corresponding optimization problem is the maximum independent set problem, which attempts to find the largest independent set in a graph.
		A much easier problem to solve is that of finding a maximal independent set, which is an independent set not contained in any other independent set.
		It fact, the independent set problem and the clique problem are equivalent, in the sense that if we know one is NP-complete, we can easily show that the other is NP-complete, and most algorithms for solving one problem can be transformed into an algorithm which solves the other in the same time and space.
	www.nationmaster.com /encyclopedia/Independent-Set-problem (1346 words)

	Subset sum problem - Wikipedia, the free encyclopedia
		The subset sum problem is an important problem in complexity theory and cryptography.
		The problem is NP-Complete, and is perhaps the simplest such problem to describe.
		It is a decision, and not an optimization problem and
	www.wikipedia.org /wiki/Subset_sum_problem (1288 words)

	Talk:NP-complete problem - Wikipedia, the free encyclopedia
		The points about P problems not necessarily being easy to solve are all well taken, but should probably be moved to the article that describes the two classes P and NP.
		For the second bullet, I said NP-complete problems are the hardest in NP in the first sentence of my article, but wanted to give some caveats to that near the end.
		They are even more confused when they learn that there may be an NP problem that is not in NP-complete, but that is more difficult than a given NP-complete problem by a factor of O(n^1000).
	en.wikipedia.org /wiki/Talk:NP-complete_problem (363 words)

	PSPACE-complete - Wikipedia, the free encyclopedia
		A decision problem is in PSPACE-complete if it is in PSPACE, and every problem in PSPACE can be reduced to it in polynomial time.
		These problems are widely suspected to be outside of P and NP, but that is not known.
		Another PSPACE-complete problem is the problem of deciding whether a given string is a member of the language defined by a given context-sensitive grammar.
	www.wikipedia.com /wiki/PSPACE-Complete (398 words)

	NationMaster - Encyclopedia: Boolean satisfiability problem
		The propositional satisfiability problem (SAT), which decides whether a given propositional formula is satisfiable, is of central importance in various areas of computer science, including theoretical computer science, algorithmics, artificial intelligence, hardware design and verification.
		This problem is solved by the polynomial-time Horn-satisfiability algorithm, and is in fact P-complete.
		Although this problem seems easier, it has been shown that if there is a practical (randomized polynomial-time) algorithm to solve this problem, then all problems in NP can be solved just as easily.
	www.nationmaster.com /encyclopedia/Boolean-satisfiability-problem (3005 words)

	Clay Mathematics Institute
		A problem is of type P if it can be solved using an algorithm whose running time grows no faster than some fixed power of the number of symbols required to specify the initial data.
		The SAT problem for a given Boolean circuit can be 'encoded' as a Minesweeper Consistency Problem for some position in the game, using a code procedure that runs in polynomial time.
		The upshot is that solving the Minesweeper Consistency Problem is algorithmically equivalent to the SAT problem, and is thus NP-complete.
	www.claymath.org /Popular_Lectures/Minesweeper (2085 words)

	NP-complete Information - TextSheet.com (Site not responding. Last check: 2007-10-21)
		In complexity theory, the NP-complete problems are the hardest problems in NP, in the sense that they are the ones most likely not to be in P.
		One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them adds up to zero.
		An interesting example is the problem, in graph theory, of graph isomorphism.
	www.rsk.soldat.sferahost.com /encyclopedia/n/np/np_complete_1.html (1115 words)

	Complete Problems
		If problem A is reduced to problem B by a function in the set R and A is hard relative to R, then B must be hard relative to R as well.
		A problem Q is complete for C under R-reduction if it is hard for C under R-reductions and is a member of C. Problems are hard for a class if they are as hard to solve as any other problem in the class.
		Thus, complete problems are the hardest problems in the class.
	www.geocities.com /s2swen/song.html (1657 words)

	NP-Completeness (Site not responding. Last check: 2007-10-21)
		Problems that can be solved using a reasonable amount of memory (again defined formally as a polynomial in the input size) without regard to how much time the solution takes.
		So this is one of the rare examples of a problem that can often be solved efficiently in practice even though it is theoretically not known to be in P. Certain related problems in higher dimensions (is this four-dimensional surface equivalent to a four-dimensional sphere) are provably undecidable.
		This problem takes as input a program X and a number K. The problem is to find data which, when given as input to X, causes it to stop in at most K steps.
	www.ics.uci.edu /~eppstein/161/960312.html (3273 words)

	NP-complete Summary
		Problems that have a solution that runs in polynomial time are in the class P. Problems that are in NP have algorithms that run in exponential or factorial time.
		One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them sums to zero.
		John Hopcroft brought everyone at the conference to a consensus that the question of whether NP-complete problems are 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.
	www.bookrags.com /NP-complete (2196 words)

	UW-Waukesha - Math Problem Solving
		Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation.
		Sometimes the "applied" problems don't appear very realistic, but that's usually because the corresponding real applied problems are too hard or complicated to solve at your current level.
		Then complete the conversion of the problem into math, i.e., find equations which describe relationships among the variables, and describe the goal of the problem mathematically.
	www.waukesha.uwc.edu /trio/math_problems.html (592 words)

	Resource Desk
		Complete a problem from the Chapter 11 problems at the end of the chapter.
		Complete a problem from the Chapter 12 problems at the end of the chapter.
		Complete a problem from the Chapter 13 problems at the end of the chapter.
	www.dacc.cc.il.us /~lhicks/cacc166/costactivities03m.htm (360 words)

	Richard Kaye's minesweeper page
		Problems like this are of great practical importance, and an efficient algorithm would be very useful.
		This is one of the biggest and most important open problem in mathematics at the moment, and is the subject of a $1,000,000 prize offered by the Clay Mathematics Institute in the USA.
		For the current discussion, it suffices that the problem of simply detecting which squares are or are not mines is equivalent to the Minesweeper Consistency Problem, and the fact that it is NP-complete means, for Minesweeper fans, that their favourite game can be seen as being right at the cutting edge of mathematical research.
	for.mat.bham.ac.uk /R.W.Kaye/minesw/ordmsw.htm (1376 words)

	PSPACE-complete (Site not responding. Last check: 2007-10-21)
		problems are widely suspected to be outside P and NP but that is not known.
		This is the problem of there are assignments of truth values to that make a boolean expression true.
		Another PSPACE-complete problem is the problem of whether a given string is a member the language defined by a given context-sensitive grammar.
	www.freeglossary.com /PSPACE-Complete (379 words)

	Resource Desk
		Complete problem P2 from the Chapter 15 problems at the end of the chapter.
		Complete problem P8 from the Chapter 16 problems at the end of the chapter.
		Complete problem P 6, SD5 from the Chapter 25 problems at the end of the chapter.
	www.dacc.cc.il.us /~lhicks/Managerial/manactivities.htm (268 words)

	Co NP (Site not responding. Last check: 2007-10-21)
		Co NP Co NP In the complexity theory,co-NP is the complexity class that contains the complements of decision problems in the complexity class NP.
		The complement of a decision problem is here defined as the problem with the yes andno answers reversed, or if we define decision problems as sets of finite strings as the complement of this set withrespect to the given alphabet.
		Assume that there is an NP-complete problem that is in co-NP.Since all problems in NP can be reduced to this problem it follows that for all problems in NPwe 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.
	www.therfcc.org /co-np-45966.html (348 words)

	Co-NP-complete (Site not responding. Last check: 2007-10-21)
		In complexity theory the complexity class Co-NP -complete is the set of problems that the hardest problems in Co-NP in the sense that they are ones most likely not to be in P.
		A more formal definition: A decision problem C is Co-NP -complete if it is in Co-NP and if every problem in Co-NP is many-one reducible to it.
		This means that for Co-NP problem L there exists a polynomial time algorithm can transform any instance of L into an instance of C with the same truth value.
	www.freeglossary.com /Co-NP-complete (153 words)

	A Complete Problem for Statistical Zero Knowledge (Site not responding. Last check: 2007-10-21)
		We present the first complete problem for SZK, the class of (promise) problems possessing statistical zero-knowledge proofs (against an honest verifier).
		The problem, called STATISTICAL DIFFERENCE, is to decide whether two efficiently samplable distributions are either statistically close or far apart.
		We propose the use of complete problems to unify and extend the study of statistical zero knowledge.
	www.eecs.harvard.edu /~salil/papers/complete-abs.html (257 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The NP-hardness of the satisfiability problem was demonstrated by exhibiting the existence of a polynomial time reduction, from each problem in NP to the satisfiability problem.
		For instance, in the case of the satisfiability problem, an exhaustive search is made for an assignment to the variables that satisfies the given expression.
		In the case of the clique problem, the exhaustive search is made for a clique of the desired size.
	www.cse.ohio-state.edu /~gurari/theory-bk/theory-bk-fivese4.html (1241 words)

	Traveling Salesman Problem
		Problem description: Find the cycle of minimum cost that visits each of the vertices of G exactly once.
		This path must be a single edge if the graph is complete and obeys the triangle inequality, as with points in the plane.
		Algorithm 608 [Wes83] of the Collected Algorithms of the ACM is a Fortran implementation of a heuristic for the quadratic assignment problem, a more general problem that includes the traveling salesman as a special case.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE175.HTM (1630 words)

	Math Forum: Rubric - Coding Pre-Algebra PoW Problem Difficulty
		Coding of problem difficulty focuses on the mathematical challenges represented by the problem, the difficulty of the mathematical concept, and the difficulty of mathematical calculations for students at a given level of problem solving.
		This problem could be solved by the guess-and-check method, which should be prior knowledge for the student and would ordinarily cause this to be scored as a level 1 problem.
		The concept behind this problem is easily within the range of ability for students in this grade band, but the problem contains smaller components that students can overlook or answer incorrectly; for example reporting the cost of individual tiles rather than the cost of a complete box of tiles.
	mathforum.org /library/problems/sets/prealg_difficulty.html (746 words)

	Complexity Theory - NP and NP-completeness
		To solve a decision problem `all' that is needed is an algorithm to determine if there is a genuine witness, W is in Wn for any given input instance I of size n.
		For example if T is a decision problem on numbers and S is one on graphs, then transform given a number would construct a graph using that number.
		A proof that a decision problem is NP-complete is accepted as evidence that the problem is intractable since a fast method of solving a single NP-complete problem would immediately give fast algorithms for all NP-complete problems.
	www.csc.liv.ac.uk /~ped/teachadmin/algor/npcomp.html (1801 words)

	RSA Laboratories - 2.3.1 What is a hard problem?
		Public-key cryptosystems (see Question 2.1.1) are based on a problem that is in some sense difficult to solve.
		Put simply, a problem is in P if it can be solved in polynomial time (see Section A.7), while a problem is in NP if the validity of a proposed solution can be checked in polynomial time.
		However, it is not known whether this problem is in P. The question of whether or not P = NP is one of the most important unsolved problems in all of mathematics and computer science.
	www.rsa.com /rsalabs/node.asp?id=2187 (408 words)

	Success in Mathematics
		Sometimes the "applied" problems don't appear very realistic, but that's usually because the corresponding real applied problems are too hard or complicated to solve at your current level.
		In the book each problem appears at the end of the section in which you learned how do to that problem; on a test the problems from different sections are all together.
		When you work on a harder problem, spend the allotted time (e.g., 5 minutes) on that question, and if you have not almost finished it, go on to another problem.
	euler.slu.edu /Dept/SuccessinMath.html (2317 words)

	More NP-Complete Problems
		Problems in graph theory are always interesting, and seem to pop up in lots of application areas in computing.
		One of the neat things about graph problems is that asking a question about a graph is often equivalent to asking quite a different one about the graph's complement.
		Consider the next problem which inquires as to how many vertices must be in any set which is connected to or covers all of the edges.
	www.cs.uky.edu /~lewis/cs-heuristic/text/class/more-np.html (2826 words)

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