What is Complete: Definition and 404 Discussions

In computational complexity theory, a problem is NP-complete when:

a brute-force search algorithm can solve it, and the correctness of each solution can be verified quickly, and
the problem can be used to simulate any other problem with similar solvability.The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a deterministic algorithm to check a single solution, or for a nondeterministic Turing machine to perform the whole search. "Complete" refers to the property of being able to simulate everything in the same complexity class.
More precisely, each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested quickly (in polynomial time), such that the output for any input is "yes" if the solution set is non-empty and "no" if it is empty. The complexity class of problems of this form is called NP, an abbreviation for "nondeterministic polynomial time". A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it even though it may not be in NP. Conversely, 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.
Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly. That is, the time required to solve the problem using any currently known algorithm increases rapidly as the size of the problem grows. As a consequence, determining whether it is possible to solve these problems quickly, called the P versus NP problem, is one of the fundamental unsolved problems in computer science today.
While a method for computing the solutions to NP-complete problems quickly remains undiscovered, computer scientists and programmers still frequently encounter NP-complete problems. NP-complete problems are often addressed by using heuristic methods and approximation algorithms.

View More On Wikipedia.org
Recent contents Top users
  1. bleist88

    Obtaining a Complete Position function in complex system

    Hi, {First off I should share my ultimate goal, which is to have a position function of time of several bodies, let's say in a solar system, so that the position of each body and the acceleration due to gravity on all the other bodies in the system are given at any time t. Although Kepler's...
  2. bleist88

    Complete position function of time for multibody systems and gravity (Newtonian)

    Hi, {First off I should share my ultimate goal, which is to have a position function of time of several bodies, let's say in a solar system, so that the position of each body and the acceleration due to gravity on all the other bodies in the system are given at any time t. Although Kepler's...
  3. K

    Quantum Gravity Theories: What is UV Complete?

    What theories of quantum gravity are currently UV complete?
  4. A

    What is a Complete Metric Space in Mathematics?

    Can someone help me understand the notion of complete metric space? I've read the definition (the one involving metrics that go to 0), but I can't really picture what it is. Does anyone have any examples that could help me understand this?
  5. Q

    Hydrogen atom - complete orthornormal set

    Hi, I was wondering if the bound solutions to the radial part of the hydrogen atom form a complete set for the functions in L^2(0,\infty). I know that the laguerre polynomials are complete and that they only differ from the radial solutions by factors of x^l * exp, so I thought that they would...
  6. J

    Should I stay and complete a physics degree on top of my math degree?

    My situation is different than most people who come to this board; I think anyways. I'm currently in my 4th year of school and can complete my mathematics degree next May. I can also choose to stay one more year and complete a physics degree on top of the math degree. However here's where I...
  7. X

    How does it complete the loop?

    I know that current can flow in loop only. I mean circuit must be complete for current to flow. So my question is: If someone is hanging in the air holding power transmission line, he doesn't get shock. Because circuit is incomplete. But if he touches ground in same position then he receives...
  8. A

    Complete subspace of function space

    Homework Statement (This is for a functional analysis course.) Let C(X,Y) be the space of continuous functions X \to Y. Let \mathcal{C}_0 = \{\varphi \in C([0,\infty),\mathbb{R}) : lim_{x\to \infty}\varphi(x) = 0 \}. Let \mathcal{C}_0^1 = \{\varphi \in \mathcal{C}_0 : \varphi' \in...
  9. M

    Hilbert space question; show Y is complete iff closed

    I would like a second opinion on my answer to this question as I'm confusing myself thinking about my proof. Any input is appreciated Homework Statement "Let (X, ||.||) be a complete normed linear space and Y \subsetX be a non-empty subspace of X. Then (Y, ||.||) is a normed linear...
  10. M

    Hilbert space question; show Y is complete iff closed

    I would like a second opinion on my answer to this question as I'm confusing myself thinking about my proof. Any input is appreciated Homework Statement "Let (X, ||.||) be a complete normed linear space and Y \subsetX be a non-empty subspace of X. Then (Y, ||.||) is a normed linear space...
  11. B

    Why is completeness important in characterizing ordered fields?

    This is not a homework problem. I came across this in an analysis book: In a complete ordered field (not specifically R) a member that is not zero is positive ⇔ this member is a square. WHY? How can we prove it? Is this similar to Hilbert's 17th problem?
  12. C

    What is meant by no complete TOE ?

    What is meant by "no complete TOE"? What does it mean exactly when people say "there is currently no complete quantum theory of the remaining fundamental force, gravity" for a Theory of Everything. I know there are a few competing theories. Spin Foam, String Theory, etc... Are those not...
  13. P

    Complete Dissociation: Understand Strong Acids & Bases

    This is probably an easy question, but I can't get it straight and searching has not helped. So when salts go into solution they completely dissociate if they contain anions of strong acids or cations of strong bases, right? I'm still shaky on this concept though. Take Sodium carbonate for...
  14. N

    Does an Extra ½ Year in Completing a Degree Affect Your Academic Future?

    Hi I guess it is somewhat standard to complete a M.Sc. in 2 years. But let's hypothetically say that a students spends 2½ years instead completing the degree, but still obtains a high average. Will the fact that an extra ½ year was spent be seen as something negative in academia when applying...
  15. J

    Is this a complete undergrad pure math curriculum?

    I was notified that these are all the proof-based pure math classes offered at my university for the undergrad pure math degree: Linear Algebra Advanced Calculus Foundations of Geometry Elementary Number Theory Complex Analysis Abstract Algebra Since the essence of pure math is...
  16. E

    Spring compression to complete a loop (energy)

    Homework Statement An 840kg roller-coaster car is launched from a giant spring of constant k=31kN/m into a frictionless loop-the-loop track of radius 6.2m. What is the minimum amount that the spring must be compressed if the car is to stay on the track? Homework Equations Wspring=0.5kx^2...
  17. P

    Can we prove that there is a maximal complete set of commuting

    Hi, Just wondering, can we prove that there is a maximal complete set of commuting observable?
  18. Z

    Is the space of total angular momentum complete?

    Hi It's easy to see that for addition of 2 angular momenta l1 and l2 , the space l1 m1 , l2 m2 is equivalent to the space of l1 l2 l m (where l is the total angular momentum). Counting the total number of states is usually a convenient way to make sure you got the addition right. But what...
  19. T

    Which physics texts have COMPLETE solutions manuals

    I've been reading halliday and resnick, but i have no motivation to do the problems since they only have solutions for a small minority of them. i was wondering if anyone knows of calc based texts with a corresponding complete solutions manual (or at least every odd/even problem at the very...
  20. S

    Why is there bradycardia in complete heart block?

    Ok I know that atria and ventricles contract independently and it makes sense heart would beat slower but what is the exact reason. The accesssory pacemaker that activates ventricles, is is lot slower than AV node is that the reason? Thanks :smile:
  21. M

    I proving l^2 is a complete metric space

    Homework Statement Prove that the sequence space l^2 (the set of all square-summable sequences) is complete in the usual l^2 distance. Homework Equations No equations.. just the definition of completeness and l^2. The Attempt at a Solution I have a sample proof from class to show...
  22. E

    Complete list of math branches

    I want to compile a list of every branch of mathematics, for my studies. This is what I got from wiki: 1. Basic mathematics 2. Advanced mathematics 2.1 Pure mathematics 2.1.1 Algebra 2.1.2 Calculus and analysis 2.1.3 Geometry and topology 2.1.4 Combinatorics 2.1.5 Logic...
  23. radou

    Topologically complete space in the product topology

    Homework Statement One needs to show that a countable product of topologically complete spaces is topologically complete in the product topology. The Attempt at a Solution A space X is topologically complete if there exists a metric for the topology of X relative to which X is...
  24. S

    Minimum degree of polynomial time NP complete problem algorithm

    So no one is quite sure that P != NP, although they tend to favor that relation. But I was curious, has anyone proved a minimum degree order to any algorithm that solves NP complete problems in polynomial time? In other words, they don't know if it can be done in polynomial time, but do they...
  25. M

    Complete Works of Major Mathematicians (English)

    I am very interested in reading the original works of some famous mathematicians, such as: Cauchy Lagrange Euler Gauss The Bernoullis Riemann etc. I have found the their respective works have been collected in volumes (for example, Cauchy has 27 volumes!) but they are all in their...
  26. D

    Help me complete my knowledge

    Hello, this is my first post. Anyway, when I was in high school I did not have my priorities in order so I missed out on my math education. About a year ago or more, I started reading a brief history of time. Now today I have read all the hawking books as well as several that covered...
  27. F

    Prove that eigenstates of hermitian operator form a complete set

    Not really sure how to go about this. Our lecture said "it can be shown" but didn't go into any detail as apparently the proof is quite long. I'd really appreciate it if someone could show me how this is done. Thanks. (Not sure if this is relevant but I have not yet studied Hilbert spaces).
  28. J

    Powersing Two 12V Motors: A Complete Guide

    Hi, new to the whole electrical side of things, have never been good with voltages, currents or resistance etc. So I apologise for any simple questions. So if I have two 12v motors, does this mean I have to include two 12v (or one 24v) supplies? Or somehow does a 12v supply power both of them...
  29. G

    Photonic band gaps: Incomplete vs complete

    Hi all, I'm currently trying to learn more about photonic band gaps. The material I am looking at is an FCC crystal of spheres with an (111) surface, and this structure is theoretically predicted to display an incomplete band gap from the gamma to L points. (Please see attachment)...
  30. xortan

    Calculators Resistor Calculator - C - Complete n00b

    Hi all, I am studying to be an electrical engineer and I have a C programming course that all of a sudden took a massive jump..last week we were writing single For statements and things of that manner and now he assigned a program that's just got my completely stumped, I thought I was...
  31. S

    Can the Archimedean Property Prove 1/n < x for All Positive Reals?

    Homework Statement Use the Archimedean property: For all x in the positive reals there exists an n in the naturals such that n > x. to prove: For all x in the positive reals there exists an n in the naturals such that 1/n < x. The Attempt at a Solution Proof: Multiplying through the...
  32. L

    What is the real power behind the rush to fill in complete emptyness

    is there something going on when i vacuum is crated from nothing at all in the atmospher? i mean why such a big rush to fill in that gap which it seems like the power to fill it in is endless and will always happen . if something can't be replaced then that something can't be removed as...
  33. P

    Complete Radioactive Decay Equations and Identify Types

    Homework Statement complete the following decay equations by inserting the missing particle or nuclide information. Identify each type of decay. see attached screenshot please. Homework Equations The Attempt at a Solution I've done all of the decays but we have to label them and...
  34. Z

    Programs Is it possible to complete a physics bachelor degree in 2 years only ?

    Is it possible in any USA universities to complete a bachelor degree in only 2 years by passing all the examinations required for graduation with attending laboratory classes only for some one who have actually Self studied physics before and is very good in it?
  35. T

    Show that the field of rational functions is not a complete ordered field

    Homework Statement Show that R(x) cannot be made into a complete ordered field, where R(x) is the field of rational functions. Homework Equations Definition of a complete ordered field: An ordered field O is called complete if supS exists for every non empty subset S of O that is...
  36. L

    Finding the excess after the reaction is complete (chem 101)

    When 2.0mL of 1.5M Ammonium Phosphate is mixed with 1.00 mL of 0.75 M Ferrous Acetate. The problem itself is in multiple parts. I have already found the limited reactant. But i need help finding "the grams of the reactant in excess after the reaction is complete." What i have so...
  37. ╔(σ_σ)╝

    Is F a Complete Ordered Field?

    Homework Statement Let F be an ordered field in which every strictly monotone increasing sequence bounded above converges. Prove that F is complete Homework Equations Definitions: Monotone Sequence property: Let F be an ordered field. We say that F has the monotone sequence...
  38. G

    Need Help with Complete System Restore on HP Desktop with Vista?

    I have an HP desktop running vista. I am wanting to do a complete system restore to restore back to original settings. I can only find where to restore to a prev. back-up. How do I go about this--HELP
  39. E

    What are factors and multiples?

    I'm learning Algebra (the basic and foundation of Math) 'cause I stink. Now I'm just doing some review from the beginning before I get into the tough stuff. I'm a bit stumped on factors and multiples. I understand it but not clear enough. I couldn't give you a definition and if you asked a...
  40. J

    Proof that gamma matrices form a complete basis

    Hi all, I'm interested in proving/demonstrating/understanding why the Dirac gamma matrices, plus the associated tensor and identity, form a complete basis for 4\times4 matrices. In my basic QFT course, the Dirac matrices were introduced via the Dirac equation, and we proved various...
  41. D

    The Subset Multiply Problem: Is it NP Complete?

    The subset sum problem is NP complete. What if we replace summing with multiplying? Would it still be np complete?
  42. P

    Complete the chemical equation

    Homework Statement Complete and balance the following chemical equation BaO + P2O5 Homework Equations The Attempt at a Solution The only one possible product I've been able to arrive at is Ba(PO3)2 but this is based on no science whatsoever. I can't find anything in my text...
  43. C

    Revisiting the Completeness of Quantum Theory: A Scientist's Perspective

    I am sorry if this issue has been already addressed previously in this forums, but I have been looking for old threads on the subject and I haven't found any specific to the matter. If anyone knows about any, please let me know and sorry again. Otherwise, this is my question: "Is Quantum...
  44. J

    Complete Induction: Proving using WOP/Induction

    Homework Statement Prove the principle of complete induction using either the well-ordering principle or ordinary mathematical induction. Homework Equations N/A The Attempt at a Solution Suppose that A is a collection of natural numbers such that 1 \in A and n+1 \in A whenever 1...
  45. S

    Complete photonic view like the CMBR version

    Ignoring what we know. I was curious as to if anyone had overlayed all of the spectrum's of light on top of each other in a complete photonic view like the CMBR version, only with a much more varied scale, that being the entire light spectrum. And if so what did this produce? I only ask because...
  46. S

    Complete vaporization time of water into superheated steam - DESUPERHEATER

    Hi to all...i've a problem. i have to calculate the position and the time of complete evaporation of a continuous water flow which is injected trough a nozzle into a continuous flow of steam in superheated condition. The position (and than the time) at which I'm sure that there aren't water...
  47. B

    Complete list of physics equations?

    This may be somewhat of an odd request, but I need a list of all physics formulas in the broad subject of physics (probably just into to physics level), preferably in plaintext. Anyone know where I can get this? I'm not looking to study it, so I'm not looking for a list of nicely categorized...
  48. N

    Complete Analysis of Rotating 2-arm structure

    Hey Guys, I have a dynamics problem I've been working on for a few days and I'm not quite sure that the results I've gotten are correct, or even if the way I'm approaching it is. The problem consists of 2 arms connected to each other by a movable pivot. However, the whole structure cannot...
  49. I

    Complex gaussian, complete the square

    Homework Statement Evaluate this integral (in essence the Fourier transform of the Gaussian): \int_{-\infty}^{+\infty} e^{-ax^2} e^{ikx}\, dx 2. The attempt at a solution One way is to complete the square, so that -ax^2 + ikx = -a(x-ik/2a)^2 - (k/2a)^2 so the integral becomes...
  50. Somefantastik

    Complete, Equivalent, Closed sets

    If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete? What if it is known that A is closed, can it then be said that B is also closed?
Back
Top