Computational problems in graph theory | NP-complete problems
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size. The clique problem arises in the following real-world setting. Consider a social network, where the graph's vertices represent people, and the graph's edges represent mutual acquaintance. Then a clique represents a subset of people who all know each other, and algorithms for finding cliques can be used to discover these groups of mutual friends. Along with its applications in social networks, the clique problem also has many applications in bioinformatics, and computational chemistry. Most versions of the clique problem are hard. The clique decision problem is NP-complete (one of Karp's 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques. Therefore, much of the theory about the clique problem is devoted to identifying special types of graph that admit more efficient algorithms, or to establishing the computational difficulty of the general problem in various models of computation. To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming to be practical for networks comprising more than a few dozen vertices.Although no polynomial time algorithm is known for this problem, more efficient algorithms than the brute-force search are known. For instance, the Bron–Kerbosch algorithm can be used to list all maximal cliques in worst-case optimal time, and it is also possible to list them in polynomial time per clique. (Wikipedia).
Clique Problem - 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
What is a Clique? | Graph Theory, Cliques
What is a clique? A clique in graph theory is an interesting concept with a lot of depth to explore. We define the term and give some examples in today's math video lesson! A clique C of a graph G is usually defined as a subset of the vertex set of G such that every pair of distinct verti
From playlist Graph Theory
General Clique - 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
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
What is a Maximal Clique? | Graph Theory, Cliques, Maximal Cliques
What are maximal cliques? Cliques are cool, but there are some special types of cliques as well, one of which is the maximal clique, which we discuss in today's graph theory video lesson! Remember that a clique is a set of vertices that are all adjacent in a graph, or it is also sometimes
From playlist Graph Theory
Maximum and Maximal Cliques | Graph Theory, Clique Number
What are maximum cliques and maximal cliques in graph theory? We'll be defining both terms in today's video graph theory lesson, as well as going over an example of finding maximal and maximum cliques in a graph. These two terms can be a little confusing, so let's dig in and clarify our un
From playlist Graph Theory
Clojure - the Reader and Evaluator (4/4)
Part of a series teaching the Clojure language. For other programming topics, visit http://codeschool.org
From playlist the Clojure language
Part of a series teaching the Clojure language. For other programming topics, visit http://codeschool.org
From playlist the Clojure language
CSE 373 -- Lecture 25, Fall 2020
From playlist CSE 373 -- Fall 2020
What are the Maximum and Maximal Cliques of this Graph? | Graph Theory
How do we find the maximum and maximal cliques of a graph? We'll go over an example in today's graph theory lesson of doing just that! To use these elementary methods, we just need to remember our definitions. A clique of a graph G is a complete subgraph of G. We also call the vertex set
From playlist Graph Theory
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
MIT 6.046J Design and Analysis of Algorithms, Spring 2015 View the complete course: http://ocw.mit.edu/6-046JS15 Instructor: Amartya Shankha Biswas In this recitation, problems related to NP-Completeness are discussed. License: Creative Commons BY-NC-SA More information at http://ocw.mit
From playlist MIT 6.046J Design and Analysis of Algorithms, Spring 2015
This is Lecture 22 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/lecture24.pdf
From playlist CSE373 - Analysis of Algorithms - 1997 SBU
Network Analysis. Lecture 8. Network communitites
Cohesive subgroups. Graph cliques, k-plexes, k-cores. Network communities. Vertex similarity matrix. Similarity based clustering. Agglomerative clustering. Graph partitioning. Repeated bisection. Edge Betweenness. Newman-Girvin algorithm. Lecture slides: http//www.leonidzhukov.net/hse/201
From playlist Structural Analysis and Visualization of Networks.
This is Lecture 25 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/lecture21.pdf More inf
From playlist CSE373 - Analysis of Algorithms 2016 SBU
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
Clojure - the Reader and Evaluator (2/4)
Part of a series teaching the Clojure language. For other programming topics, visit http://codeschool.org
From playlist the Clojure language