NP-complete problem

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Topic: NP-complete problem

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	Complete Problems
		The proof for each problem is simply to reduce a problem previously shown NP-Complete to the current problem.
		Thus, complete problems are the hardest problems in the class.
		Complete problems are members of the class for which they are hard.
	www.geocities.com /s2swen/song.html (1657 words)

	NP-complete
		Informally, a problem is NP-complete if answers can be verified quickly, and a quick algorithm to solve this problem can be used to solve all other NP problems quickly.
		Eppstein's longer, but very good introduction to NP-completeness, with sections like Why should we care?, Examples of problems in different classes, and how to prove a problem is NP-complete.
		Definition: The complexity class of decision problems for which answers can be checked for correctness, given a certificate, by an algorithm whose run time is polynomial in the size of the input (that is, it is NP) and no other NP problem is more than a polynomial factor harder.
	www.nist.gov /dads/HTML/npcomplete.html (281 words)

	NP-Complete Problems
		To prove that a problem P is NP-complete:
		Here is a proof of the NP-Completeness of the CNF Satisfiability problem by Dr. Dunne.
		NP problems thus deal with decision problems as opposed to optimization problems.
	dcm.cl.uh.edu /yue/courses/csci5432/npcomplete.asp (742 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)

	Stas Busygin's NP-Completeness Page
		Hence complete problems are hardest in their own classes and as they exist we may choose any of them to advance solving techniques for the entire class.
		The concept of complete problems for a class is generalized to hard problems for the class by inclusion of all other problems, whose polynomial time algorithm gives polynomial time solvability for the class.
		A problem of a class is complete if you can solve any other problem of this class in polynomial time having a polynomial time algorithm for the first one.
	www.busygin.dp.ua /npc.html (896 words)

	SAT
		Say we have an NP-complete problem XYZ whose raw input typically is quite compressible in polynomial time by a function f, and where the decompression is polynomial time in the number of bits in the compressed string for the function (algorithm) g.
		It is hard to pin down more closely, because the encoding of the problem affects the parameter--if we can compress the input by a factor of b, we effectively multiply k by b (or raise alpha to the b'th power) using the same algorithm (plus a subexponential-time decompression step).
		Problems on planar graphs, or two-dimensional geometric problems, such as the planar or geometric TSP problems, can often be solved in time exponential in O(sqrt n) or O(sqrt n log n) instead of exponential in n by using a separator-based divide and conquer.
	www.math.niu.edu /~rusin/known-math/00_incoming/SAT (2344 words)

	NP-complete Problems
		NP-complete problems are a class of problems that exhibit exponential growth in all current algorithms.
	www.shodor.org /scsi/handouts/brachi/sld004.htm (28 words)

	Program Optimisation Catalogue - NP, NP-Complete And NP-Hard Problems
		An NP-hard problem, is the same as an NP-complete problem, in that a solution in polynomial time would lead to a solution to all problems in NP in polynomial time.
		An NP-complete problem is a problem in NP, for which, if there existed a polynomial time algorithm, would prove that all problems in NP had a polynomial time solution.
		Problems in NP with no known polynomial time solution are not necessarily NP-complete.
	users.chariot.net.au /~matty/optcat/np_problems.html (238 words)

	Some Well-known NP-Complete Problems
		If any NP-complete language is in the class P, then P = NP.
		A 0-1 problem P is considered to be a language L by encoding each problem instance as a string, and then letting L be the set of all encodings of such instances where the answer is 1.
		A 0-1 problem is a problem where the answer is always 0 or 1.
	www.egr.unlv.edu /~larmore/Courses/CSC456/npcomplete.html (366 words)

	PCGrate -> NP-Complete Problems
		At the present time, the question of existence of algorithms which determine "the proper line" for solving NP-complete problems in polynomial time is still an open question.
		The theory of intractable problems may be treated as a certain universal mathematical philosophy which is applied by experts in the context of relations between certain classes of algorithms (P = NP or P ≠ NP).
		A problem which can be solved by a series of guessing (non-deterministic) steps but whose solution can be verified as correct in polynomial time is said to lie in class NP.
	www.pcgrate.com /about/npcomprb (668 words)

	CSC-105 2000S : What implications do you see in the existence of NP-complete problems?
		If a problem is NP and all other NP problems are polynomial-time reducible to it, the problem is NP-complete.
		What implications do you see in the existence of NP-complete problems?
		A problem is called NP (nondeterministic polynomial) if its solution (if one exists) can be guessed and verified in polynomial time; nondeterministic means that no particular rule is followed to make the guess.
	www.math.grin.edu /~rebelsky/CS105/Questions/question.42.html (1297 words)

	CSC 4170 NP-Complete Problems
		This is what the "complete" refers to when we talk about NP-complete problems.
		All of the known NP problems have a remarkable characteristic: they are all reducible to one another.
		What this means is that, given any two NP problems X and Y, There exists a polynomial-time algorithm to restate a problem of type X as a problem of type Y, and
	www.seas.upenn.edu /~cit596/notes/dave/p-and-np7.html (204 words)

	Introduction
		Completeness proofs of these problems are given in Sections 4, 5, and 6.
		In particular, they showed that any NP search problem with a p-samplable distribution is reducible to an NP search problem with a uniform distribution under a randomized reduction.
		Given a problem and a distribution on instances, finding an expected polynomial-time algorithm to solve the problem or proving that such an algorithm does not exist is an important issue.
	www.uncg.edu /mat/avg/avgnp/node2.html (987 words)

	Brian's Digest: NP Problems
		If P != NP then it is impossible to write a program that solves all NPC problem instances in polynomial worst case time.
		The problem can be regarded as a subset sum problem in which only the t most significant bits of the right hand side are prescribed, with t bounded by a polynomial in log n.
		The variant of the problem in which only the t _least_ significant bits are prescribed is clearly polynomially solvable by doing additions modulo 2^t.
	www.worms.ms.unimelb.edu.au /digest/np_problems.html (1239 words)

	NP-Complete Problems
		In order for a problem to be in the class NP-Complete, it must first be in the class NP.
		If a polynomial time solution is found for any of the NP-Complete problems, then every NP problem can be solved in polynomial time.
		Or, are there problems that are in the class NP that are not in the class P. No one has been able to prove that they are or are not the same class.
	www.cs.uidaho.edu /~karenv/cs101/lectures2/np.complete.html (168 words)

	Brian's Digest: NP Problems
		It is quite possible that the appropriate mapping from a particular NP complete problem to the general solution procedure would map some of the practical cases onto the intractable 'special cases'.
		Whether your problem is in NP is a different question: you have to verify whether a nondeterministic "guess" could be verified in polynomial time.
		Whether your problem is in NP is a ->different question: you have to verify whether a nondeterministic ->"guess" could be verified in polynomial time.
	www.worms.ms.unimelb.edu.au /digest/np_problems96.html (4667 words)

	charlotte.html
		NP-complete problems are considered to be the hardest problems, since if any problem in NP is shown to be intractable then all NP-complete problems are intractable.
		A problem in NP is NP-complete if every other problem in NP can be expressed in terms of it by means of a polynomial time algorithm.
		In this paper, two DNA algorithms are presented for the Road Coloring problem, a problem that is in NP but not known to be NP-complete.
	www.csd.uwo.ca /~jamie/.Refs/Courses/CS881/charlotte.html (2332 words)

	NP-complete
		So, an example of an NP problem is, "find the combination to this suitcase." If you come up with a way to answer that question, it's easy for me to tell whether you're right.
		Well, Stephen Cook proved in 1972 that a problem called Circuit-SAT is at least as hard to solve as any other NP problem.
		He didn't get any closer to a complete, consistent formulation of math, but he at least saved everyone else the trouble by proving that it couldn't be done.
	tomclegg.net /npcomplete (2620 words)

	Group Hodgson - Biocomputing - NP-complete problems
		This problem involves assigning colours to vertices in such a way that no two adjoining vertices are similarly coloured.
		This problem is composed of a graph of n vertices, linked by legal paths.
		Using DNA to solve such problems allows linear computational time because of the inherent massive parrallelism, BUT solutions require an exponential mass of DNA.
	template.bio.warwick.ac.uk /staff/dhodgson/dnacomp/np (1025 words)

	Other NP-Complete Problems
		A few other catalog problems exist in a limbo state, where it is not known whether the problem has a fast algorithm or is NP-complete.
		Clique, vertex cover, and integer programming are just three of the literally hundreds of problems that have been shown to be NP-complete.
		It is important to be aware of which kinds of problems tend to be hard, so you recognize them when you see them in applications, and also to provide a suitable class of candidates for future reductions.
	www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK3/NODE116.HTM (459 words)

	IA Publications
		The methodology is to devise network architectures which can be trained on examples from the problem domain in order to produce a network which solves the problem in general.
		For optimization problems that are computationally intractable, I also try to develop approximation algorithms with good performance guarantees.
		I am currently working on unification problems that are relevant in protocol analysis.
	www.albany.edu /cifa/publications.html (10832 words)

	NP-Complete Problems
		Given an instance of the vertex-cover problem, the reduction algorithm constructs an instance of the subset-sum problem such that G has a vertex cover of size k if and only if there is a subset of S whose sum is exactly t.
		To show that clique is in NP, for a given graph G = (V, E), we use the set V' C V of vertices in the clique as a certificate for G.
		The clique problem is the optimization problem of finding a clique of maximum size in a graph.
	www.msci.memphis.edu /~giri/7713/f99/kolli/NPComplete.html (922 words)

	NP-Complete Problems in the Art Gallery
		The first problem is called the "Art-Gallery" Problem (thus, the gallery analogy).
		This is a well known problem in computational geometry.
		Many results have been discovered with respect to this problem.
	cgm.cs.mcgill.ca /~cwu25/proj507/intro507.html (191 words)

	4.2 General Solutions to NP-complete problems
		Recall from section 1.2 that all NP-complete problems can be reduced to all other NP-complete problems.
		Alternative solutions to NP-complete problems using the DNA computing model have been expressed, and some are mentioned here.
		This suggests that all NP-complete problems can be solved using the DNA computing model.
	www.cc.gatech.edu /grads/b/bradf/cs7001/proj1/dna_comp-node13.html (105 words)

	Representing Reductions of NP-Complete Problems in (ResearchIndex)
		Furthermore, any instance of a NP-complete problem can be converted to an instance of another problem in NP in polynomial time.
		Thus, identifying NP-complete problems is very important in algorithm design and can help computer scientists and engineers redirect their e#orts towards finding approximate solutions to these problems.
		Abstract: Under the widely believed conjecture P#=NP, NP-complete problems cannot be solved exactly using e#cient polynomial time algorithms.
	citeseer.ist.psu.edu /627218.html (323 words)

	Jennifer Dodd
		The beauty of a problem like this is that I have to learn about resistance and how that depends on properties of wires and voltage etc, so the lecturer can be certain that I will gain some specific knowledge in the course of coming to a recommendation.
		These problems are mildly humorous, but they’re no more realistic or motivating than a problem stated like this: “Two bodies of mass m are sliding on a frictionless plane at a speed of …”.
		Doxiadis identifies the major problem with teaching mathematical subjects: the final form of a mathematical idea is not the form in which it is best taught.
	www.physics.uq.edu.au /people/jdodd (3373 words)

	Traveling Salesperson & NP-Complete
		Loosely speaking, NP-Complete is a class of problems that are believed unsolvable within a reasonable amount of time in the worst case.
		Computer scientists have proved that the quasigroup completion problem belongs to a category of difficult computational problems described as NP-complete.
		The so-called quasigroup completion problem concerns a table that is correctly but only partially filled in.
	www.aaai.org /AITopics/html/tsp.html (624 words)

	NP-complete problems
		if any one NP-complete problem really is in P then all are in P
		An optimization problem can usually be expressed as a polynomial sequence of decision problems and so can be said to be polynomially equivalent to the corresponding decision problem.
		A decision problem is a question, the correct answer to which is either Yes or No.
	www.ics.uci.edu /~dan/class/161/notes/10/NP.html (321 words)

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