NL-complete problems | Satisfiability problems

In computer science, 2-satisfiability, 2-SAT or just 2SAT is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable. But in contrast to those more general problems, which are NP-complete, 2-satisfiability can be solved in polynomial time. Instances of the 2-satisfiability problem are typically expressed as Boolean formulas of a special type, called conjunctive normal form (2-CNF) or Krom formulas. Alternatively, they may be expressed as a special type of directed graph, the implication graph, which expresses the variables of an instance and their negations as vertices in a graph, and constraints on pairs of variables as directed edges. Both of these kinds of inputs may be solved in linear time, either by a method based on backtracking or by using the strongly connected components of the implication graph. Resolution, a method for combining pairs of constraints to make additional valid constraints, also leads to a polynomial time solution. The 2-satisfiability problems provide one of two major subclasses of the conjunctive normal form formulas that can be solved in polynomial time; the other of the two subclasses is Horn-satisfiability. 2-satisfiability may be applied to geometry and visualization problems in which a collection of objects each have two potential locations and the goal is to find a placement for each object that avoids overlaps with other objects. Other applications include clustering data to minimize the sum of the diameters of the clusters, classroom and sports scheduling, and recovering shapes from information about their cross-sections. In computational complexity theory, 2-satisfiability provides an example of an NL-complete problem, one that can be solved non-deterministically using a logarithmic amount of storage and that is among the hardest of the problems solvable in this resource bound. The set of all solutions to a 2-satisfiability instance can be given the structure of a median graph, but counting these solutions is #P-complete and therefore not expected to have a polynomial-time solution. Random instances undergo a sharp phase transition from solvable to unsolvable instances as the ratio of constraints to variables increases past 1, a phenomenon conjectured but unproven for more complicated forms of the satisfiability problem. A computationally difficult variation of 2-satisfiability, finding a truth assignment that maximizes the number of satisfied constraints, has an approximation algorithm whose optimality depends on the unique games conjecture, and another difficult variation, finding a satisfying assignment minimizing the number of true variables, is an important test case for parameterized complexity. (Wikipedia).

What is an Injective Function? Definition and Explanation

An explanation to help understand what it means for a function to be injective, also known as one-to-one. The definition of an injection leads us to some important properties of injective functions! Subscribe to see more new math videos! Music: OcularNebula - The Lopez

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From playlist Decision Support Systems

Proof: Supremum and Infimum are Unique | Real Analysis

If a subset of the real numbers has a supremum or infimum, then they are unique! Uniqueness is a tremendously important property, so although it is almost complete trivial as far as difficulty goes in this case, we would be ill-advised to not prove these properties! In this lesson we'll be

From playlist Real Analysis

The Composition of Injective(one-to-one) Functions is Injective Proof

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Proof that the composition of injective(one-to-one) functions is also injective(one-to-one)

From playlist Proofs

3 Absolute Inequalities You Have To Know to Pass

In this video we are going to focus on three absolute value inequalities that you need to know to pass any test or quiz. These are not the hardest but also not the most difficult. We will solve and graph our solution. ⭐ Easy absolute inequalities - https://youtu.be/0ArjkZT3eIQ ⭐ Hard abs

From playlist Equations and Inequalities | Algebra 2

SAT Math (Geometry) - Practice 1.6

The video provides an explanation of a practice SAT math question involving goemetry. View or download the practice questions at http://www.ck12.org Site: http://mathispower4u.com

From playlist SAT Math Practice

How to Solve Absolute Value Inequalities

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys How to Solve Absolute Value Inequalities

From playlist College Algebra

How to Prove a Function is Injective(one-to-one) Using the Definition

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys How to prove a function is injective. Injective functions are also called one-to-one functions. This is a short video focusing on the proof.

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Maximum and Minimum of a set In this video, I define the maximum and minimum of a set, and show that they don't always exist. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh

From playlist Real Numbers

This is Lecture 21 of the CSE373 (Analysis of Algorithms) taught by Professor Steven Skiena [http://www.cs.sunysb.edu/~skiena/] at Stony Brook University in 1997. The lecture slides are available at: http://www.cs.sunysb.edu/~algorith/video-lectures/1997/lecture23.pdf

From playlist CSE373 - Analysis of Algorithms - 1997 SBU

Algebra - Ch. 31: Linear Inequality in 2 Variables (5 of 14) Steps to Solve Multiple Linear Ineq

Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn the steps to solve a multiple linear inequality problem: 1) put into y-slope intercept form (BOTH INEQUALITIES) 2) find

From playlist ALGEBRA CH 31 LINEAR INEQUALITIES IN 2 VARIABLES

This is Lecture 24 of the CSE373 (Analysis of Algorithms) course taught by Professor Steven Skiena [http://www3.cs.stonybrook.edu/~skiena/] at Stony Brook University in 2016. The lecture slides are available at: https://www.cs.stonybrook.edu/~skiena/373/newlectures/lecture20.pdf More inf

From playlist CSE373 - Analysis of Algorithms 2016 SBU

Algebra - Ch. 31: Linear Inequality in 2 Variables (3 of 14) How to Determine Which Region?

Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn how to determine which region satisfy the inequality. Next video in this series can be seen at: https://youtu.be/WxCjS

From playlist THE "HOW TO" PLAYLIST

Math 060 Linear Algebra #01 090117 Systems of Linear Equations

1.1 Introduction: systems of linear equations. Geometric interpretation of systems of linear equations. A system (3 variables, 3 equations) from the Han Dynasty. Operations that don't change the solution set. Definition of equivalent systems. Simplifying the system. Terminology: ov

From playlist Course 4: Linear Algebra (Fall 2017)

Top 25 Pattern Programs In C Language | Pattern Program In C Explained | C For Beginners|Simplilearn

This video by Simplilearn will explain the Top 25 Pattern Programs In C Language. Pattern Program In C Explained For Beginners tutorial will include star pattern programs, Number pattern programs, Alphabet pattern programs, and special pattern programs in c. C pattern programs are critic

From playlist C++ Tutorial Videos

(IC 2.8) Kraft-McMillan inequality - statement

For a given set of lengths, the Kraft-McMillan inequality is a necessary condition for the existence of a uniquely decodable code, and a sufficient condition for the existence of a prefix code. A playlist of these videos is available at: http://www.youtube.com/playlist?list=PLE125425EC83

From playlist Information theory and Coding

The Remarkable BEST-SAT Algorithm

A dive into the remarkable BEST-SAT approximation algorithm. Created as a part of SoME2: https://www.youtube.com/watch?v=hZuYICAEN9Y ------------------ Timetable: 0:00 - Introduction 2:21 - RAND-SAT 3:35 - LP-SAT 8:49 - BEST-SAT 10:11 - Outro ------------------ Source code: https://gi

From playlist Summer of Math Exposition 2 videos

On The Complexity of Circuit Satisfiability - Ramamohan Paturi

Ramamohan Paturi University of California at San Diego October 12, 2009 We present a gap theorem regarding the complexity of the circuit satisfiability problem. We prove that the success probability of deciding Circuit Satisfiability for deterministic circuits with n variables and size m

From playlist Mathematics

Prove or Disprove if the Function is Injective

Prove or Disprove if the Function is Injective If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Thank you:)

From playlist Functions, Sets, and Relations

Algebra - Ch. 29: Quadratic & Rational Inequalities (2 of 15) How to Solve a Quadratic Inequality?

Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn the 7 step-by-step method to solving a quadratic inequality. Next video in this series can be seen at: https://youtu.b

From playlist THE "HOW TO" PLAYLIST