Unsolved problems in mathematics | Unsolved problems in computer science | Mathematical optimization | Structural complexity theory | Millennium Prize Problems | Conjectures
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term quickly, used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions for which some algorithm can provide an answer in polynomial time is "P" or "class P". For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is NP, which stands for "nondeterministic polynomial time". An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If it turns out that P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time. The problem has been called the most important open problem in computer science. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution. (Wikipedia).
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
What Makes P vs. NP So Hard? (P ≠ EXPTIME, Time Hierarchy, Baker-Gill-Solovay)
There are a lot of unsolved problems in complexity theory, but there are a few things we do know. We look at the Time Hierarchy Theorem, and also why the proof techniques don't transfer to P vs NP. Created by: Cory Chang Produced by: Vivian Liu Script Editor: Justin Chen, Zachary Greenber
From playlist P vs NP
The Formal Definition of P (P vs NP)
Let’s take a deeper look at the complexity class P, and decision problems. Created by: Cory Chang Produced by: Vivian Liu Script Editor: Justin Chen, Elaine Chang, Zachary Greenberg, Alex Egan Twitter: https://twitter.com/UBehavior — P vs NP Playlist: https://www.youtube.com/playlist?l
From playlist P vs NP
The "P vs. NP" Problem: Efficient Computation....Knowledge" - Avi Wigderson
Avi Wigderson Institute for Advanced Study October 24, 2008 The "P vs. NP" problem is a central outstanding problem of computer science and mathematics. In this talk, Professor Wigderson attempts to describe its technical, scientific, and philosophical content, its status, and the implic
From playlist Mathematics
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
From playlist Math talks
NP: How Non-determinism Relates to Verifiable Proofs
There are multiple, surprisingly different, ways to think of NP problems. Let's talk about these different definitions and why they're equivalent. Created by: Cory Chang Produced by: Vivian Liu Script Editor: Justin Chen, Zachary Greenberg, Elaine Chang Twitter: https://twitter.com/UBeha
From playlist P vs NP
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
SAT is NP-Hard - 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
Introduction to the most famous unsolved problem in Computer Science. Introduction to Turing Machines, runtime of algorithms, and the classes P and NP. What would the universe look like if P=NP. History of the problem, and attempts to solve the problem. Example adapted from https://en.wiki
From playlist CS50 Seminars 2016
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
14. P and NP, SAT, Poly-Time Reducibility
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. Defined NTIME(t(n)) complexity classes
From playlist MIT 18.404J Theory of Computation, Fall 2020
22. Provably Intractable Problems, Oracles
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 exponential complexity clas
From playlist MIT 18.404J Theory of Computation, Fall 2020
Lower bounds for subgraph isomorphism – Benjamin Rossman – ICM2018
Mathematical Aspects of Computer Science Invited Lecture 14.3 Lower bounds for subgraph isomorphism Benjamin Rossman Abstract: We consider the problem of determining whether an Erdős–Rényi random graph contains a subgraph isomorphic to a fixed pattern, such as a clique or cycle of consta
From playlist Mathematical Aspects of Computer Science
NP-Complete Explained (Cook-Levin Theorem)
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
The Minimum Formula Size Problem is (ETH) Hard - Rahul Ilango
Computer Science/Discrete Mathematics Seminar I Topic: The Minimum Formula Size Problem is (ETH) Hard Speaker: Rahul Ilango Affiliation: Massachusetts Institute of Technology Date: March 7, 2022 Understanding the complexity of the Minimum Circuit Size Problem (MCSP) is a longstanding mys
From playlist Mathematics
History of Science and Technology Q&A (Apr. 21, 2021)
Stephen Wolfram hosts a live and unscripted Ask Me Anything about the history of science and technology for all ages. Originally livestreamed at: https://twitch.tv/stephen_wolfram/ Outline of Q&A 0:00 Stream starts 4:16 Stephen begins the stream 4:38 Hi Stephen. Is there some particular
From playlist Stephen Wolfram Ask Me Anything About Science & Technology
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