Complexity classes | NP-complete problems | Mathematical optimization
In computational complexity theory, a problem is NP-complete when: 1. * it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. 2. * the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. 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 some 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. (Wikipedia).
NP-Completeness - Intro to Algorithms
This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.
From playlist Introduction to Algorithms
NP-Completeness - Intro to Algorithms
This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.
From playlist Introduction to Algorithms
!!Con 2016 - My favorite NP-complete problem! By Mark Dominus
My favorite NP-complete problem! By Mark Dominus NP-complete problems are the hardest problems whose solutions can be efficiently checked for correctness. An efficient method of solving any NP-complete problem would translate directly into efficient solutions for all of them. Many famous
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From playlist Mathematical Physics II Uploads
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What makes a problem "harder" than another problem? How can we say a problem is the hardest in a complexity class? In this video, we provide a proof sketch of the Cook-Levin theorem, introducing a critical concept known as NP-completeness. Created by: Cory Chang Produced by: Vivian Liu Sc
From playlist P vs NP
Existence & Uniqueness Theorem, Ex1.5
Existence & Uniqueness Theorem for differential equations. Subscribe for more math for fun videos 👉 https://bit.ly/3o2fMNo For more calculus & differential equation tutorials, check out @justcalculus 👉 https://www.youtube.com/justcalculus To learn how to solve different types of d
From playlist Differential Equations: Existence & Uniqueness Theorem (Nagle Sect1.2)
Verifying a solution to the differential equation y''+y=tan(x)
Verifying a solution to the differential equation y''+y=tan(x) Subscribe for more math for fun videos 👉 https://bit.ly/3o2fMNo For more calculus & differential equation tutorials, check out @justcalculus 👉 https://www.youtube.com/justcalculus To learn how to solve different types of
From playlist Differential Equations: Existence & Uniqueness Theorem (Nagle Sect1.2)
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From playlist Solve Differential Equation (Particular Solution) #Integration
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Learn how to solve the particular solution of differential equations. A differential equation is an equation that relates a function with its derivatives. The solution to a differential equation involves two parts: the general solution and the particular solution. The general solution give
From playlist Solve Differential Equation (Particular Solution) #Integration
Lecture 23: Computational Complexity
MIT 6.006 Introduction to Algorithms, Fall 2011 View the complete course: http://ocw.mit.edu/6-006F11 Instructor: Erik Demaine License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.006 Introduction to Algorithms, Fall 2011
MIT 18.404J Theory of Computation, Fall 2020 Instructor: Michael Sipser View the complete course: https://ocw.mit.edu/18-404JF20 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP60_JNv2MmK3wkOt9syvfQWY Quickly reviewed last lecture. Covered NP-completeness; SAT and 3SAT;
From playlist MIT 18.404J Theory of Computation, Fall 2020
P vs. NP is one of the greatest unsolved problems. Just what is it, and why is it so important? Created by: Cory Chang Produced by: Vivian Liu Script Editor: Justin Chen, Brandon Chen, Elaine Chang, Zachary Greenberg Twitter: https://twitter.com/UBehavior — Extra Resources: hackerdashe
From playlist P vs NP
This lecture is an informal introduction to the P=NP question in computer science: are nondeterministic polynomial time problems (NP) the same as polynomial time problems (P)? We describe what these terms mean, give a brief history, and examine some of the arguments for and against this qu
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MIT 6.006 Introduction to Algorithms, Spring 2020 Instructor: Erik Demaine View the complete course: https://ocw.mit.edu/6-006S20 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63EdVPNLG3ToM6LaEUuStEY This lecture discusses computational complexity and introduces termi
From playlist MIT 6.006 Introduction to Algorithms, Spring 2020
Recitation 23: Computational Complexity
MIT 6.006 Introduction to Algorithms, Fall 2011 View the complete course: http://ocw.mit.edu/6-006F11 Instructor: Victor Costan License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.006 Introduction to Algorithms, Fall 2011
P vs. NP - The Biggest Unsolved Problem in Computer Science
Get a free audiobook and a 30-day trial of Audible (and support this channel) at http://www.audible.com/upandatom or text "upandatom" to 500 500 on your phone. Hi! I'm Jade. If you'd like to consider supporting Up and Atom, head over to my Patreon page :) https://www.patreon.com/upandat
From playlist Computer Science
17. Space Complexity, PSPACE, Savitch's Theorem
MIT 18.404J Theory of Computation, Fall 2020 Instructor: Michael Sipser View the complete course: https://ocw.mit.edu/18-404JF20 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP60_JNv2MmK3wkOt9syvfQWY Quickly reviewed last lecture. Introduced space complexity. Defined S
From playlist MIT 18.404J Theory of Computation, Fall 2020
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