Complexity classes | Strongly NP-complete problems | Computational complexity theory
In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a polynomial reduction to it; in combinatorial optimization, particularly, the phrase "strongly NP-hard" is reserved for problems that are not known to have a polynomial reduction to another strongly NP-complete problem. Normally numerical parameters to a problem are given in positional notation, so a problem of input size n might contain parameters whose size is exponential in n. If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size. Thus strong NP-completeness or NP-hardness may also be defined as the NP-completeness or NP-hardness of this unary version of the problem. For example, bin packing is strongly NP-complete while the 0-1 Knapsack problem is only weakly NP-complete. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming. From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP. However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded. Some strongly NP-complete problems may still be easy to solve on average, but it's more likely that difficult instances will be encountered in practice. (Wikipedia).
Strong Induction is a proof method that is a somewhat more general form of normal induction that let's us widen the set of claims we can prove. Our base case is not a single fact, but a list of all the facts up to a particular nth level. Then we demonstrate the (n+1)th level. Previous ex
From playlist Discrete Math (Full Course: Sets, Logic, Proofs, Probability, Graph Theory, etc)
Fundamentals of Mathematics - Lecture 12: Strong Ind, Nim, and the Fundamental Theorem of Arithmetic
course page: http://www.uvm.edu/~tdupuy/logic/Math52-Fall2017.html handouts - DZB, Emory videography - Eric Melton, UVM
From playlist Fundamentals of Mathematics
Which Incapacitating Agent is the Most Effective?
This time, we're ranking incapacitating agents! The term incapacitating agent is defined by the United States Department of Defense as: "An agent that produces temporary physiological or mental effects, or both, which will render individuals incapable of concerted effort in the performance
From playlist Chemistry Tierlists
Chemistry - Acids & Bases Fundamentals (21 of 35) What Is A Strong Acid?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain "What is a strong acid?" (A "strong" acid is "weak".)
From playlist CHEMISTRY 22 ACIDS AND BASES
Watch more videos on http://www.brightstorm.com/science/chemistry SUBSCRIBE FOR All OUR VIDEOS! https://www.youtube.com/subscription_center?add_user=brightstorm2 VISIT BRIGHTSTORM.com FOR TONS OF VIDEO TUTORIALS AND OTHER FEATURES! http://www.brightstorm.com/ LET'S CONNECT! Facebook ► h
From playlist Chemistry
Tungsten Vs. Titanium Comparison
Titanium and Tungsten are some of the strongest metals on the planet. Which material do you think is the strongest of the two? 🤔 What is the difference between Tungsten and Titanium? 🧐 Watch our video to discover the metal qualities of Tungsten and Titanium. 🔩🎥 To get the latest science
From playlist Theory to Reality
Chemistry - Acids & Bases (31 of 45) Comparing Acid Strengths Using % Concentrations
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the % ionized of a strong, weak, weaker, and very weak acid.
From playlist CHEMISTRY 22 ACIDS AND BASES
Proof by Strong Induction: If x + 1/x is an Integer Then x^n+1/x^n is an Integer
This video provides an example of proof by strong induction. mathispower4u.com
From playlist Sequences (Discrete Math)
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
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
Bruno Escoffier : Et si SAT était vraiment difficile? Quelques conséquences des hypothèses ETH et...
CONFERENCE Recording during the thematic meeting : « ALEA Days» the March 16, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker : Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathemat
From playlist Mathematical Aspects of Computer Science
NP Completeness III - More Reductions - Lecutre 17
All rights reserved for http://www.aduni.org/ Published under the Creative Commons Attribution-ShareAlike license http://creativecommons.org/licenses/by-sa/2.0/ Tutorials by Instructor: Shai Simonson. http://www.stonehill.edu/compsci/shai.htm Visit the forum at: http://www.coderisland.c
From playlist ArsDigita Algorithms by Shai Simonson
Structure vs Randomness in Complexity Theory - Rahul Santhanam
Computer Science/Discrete Mathematics Seminar I Topic: Structure vs Randomness in Complexity Theory Speaker: Rahul Santhanam Affiliation: University of Oxford Date: April 20, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Global symmetry from local information: The Graph Isomorphism Problem – László Babai – ICM2018
Combinatorics | Mathematical Aspects of Computer Science Invited Lecture 13.4 | 14.5 Global symmetry from local information: The Graph Isomorphism Problem László Babai Abstract: Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity stat
From playlist Combinatorics
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
On the axiomatisation of C_p with roots of unity - R. Rioux - Workshop 2 - CEB T1 2018
Romain Rioux (Paris) / 07.03.2018 On the axiomatisation of C_p with roots of unity. In the middle of the 90’s Tate and Voloch have proved a result concerning the sums of roots of unity with fixed coefficients. By using an adapted decomposition to understand the p-adic valuation of these
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
On some fine-grained questions in algorithms and complexity – V. Vassilevska Williams – ICM2018
Mathematical Aspects of Computer Science Invited Lecture 14.8 On some fine-grained questions in algorithms and complexity Virginia Vassilevska Williams Abstract: In recent years, a new “fine-grained” theory of computational hardness has been developed, based on “fine-grained reductions”
From playlist Mathematical Aspects of Computer Science
How Perfectionism Makes Us Ill
‘Perfectionists’ are generally held in high-esteem: praised for their self-discipline and refusal to compromise. Yet in truth, the trait is a manifestation of self-hatred - and must be overcome if we are ever to feel truly fulfilled. Sign up to our mailing list to receive 10% off your firs
From playlist SELF
How To Memorize The Strong Acids and Strong Bases
This chemistry video tutorial explains how to memorize the 7 strong acids and strong bases. Strong acids dissociate completely where as weak acids dissociate partially. Strong Bases are usually soluble compounds that release hydroxide ions into solution. My Website: https://www.video-
From playlist New AP & General Chemistry Video Playlist
Oracle Separation of Quantum Polynomial time and the Polynomial Hierarchy - Avishay Tal
Computer Science/Discrete Mathematics Seminar I Topic: Oracle Separation of Quantum Polynomial time and the Polynomial Hierarchy Speaker: Avishay Tal Affiliation: University of California, Berkeley Date: Oct 1, 2018 For more video please visit http://video.ias.edu
From playlist Mathematics