Computational problems in graph theory | Combinatorial optimization
Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the max-flow min-cut theorem, determining the minimum cut over a graph representing a flow network is equivalent to computing the maximum flow over the network. Given a pseudo-Boolean function , if it is possible to construct a flow network with positive weights such that * each cut of the network can be mapped to an assignment of variables to (and vice versa), and * the cost of equals (up to an additive constant) then it is possible to find the global optimum of in polynomial time by computing a minimum cut of the graph. The mapping between cuts and variable assignments is done by representing each variable with one node in the graph and, given a cut, each variable will have a value of 0 if the corresponding node belongs to the component connected to the source, or 1 if it belong to the component connected to the sink. Not all pseudo-Boolean functions can be represented by a flow network, and in the general case the global optimization problem is NP-hard. There exist sufficient conditions to characterise families of functions that can be optimised through graph cuts, such as submodular quadratic functions. Graph cut optimization can be extended to functions of discrete variables with a finite number of values, that can be approached with iterative algorithms with strong optimality properties, computing one graph cut at each iteration. Graph cut optimization is an important tool for inference over graphical models such as Markov random fields or conditional random fields, and it has applications in computer vision problems such as image segmentation, denoising, registration and stereo matching. (Wikipedia).
Vertex Cuts in Graphs (and a bit on Connectivity) | Graph Theory, Vertex-Connectivity
What is a vertex cut of a graph? And how can we use vertex cuts to describe how connected a graph is? We have discussed cut vertices and connected graphs before, but by tying them together in a way, we are able to characterize different levels of connectivity in graphs. The focus of this l
From playlist Graph Theory
Graph Theory: 53. Cut-Vertices
Here we introduce the term cut-vertex and show a few examples where we find the cut-vertices of graphs. We then go through a proof of a characterisation of cut-vertices: a vertex v is a cut-vertex if and only if there exist vertices u and w (distinct from v) such that v lies on every u-w
From playlist Graph Theory part-9
Edge Cuts and Edge Connectivity | Graph Theory
Edge cuts, minimum edge cuts, minimal edge cuts, and edge connectivity are all introduced in today's graph theory lesson! Edge cuts are similar to vertex cuts but, of course, with edges! An edge cut of a nontrivial graph G is a set, X, of edges of G, such that G-X is disconnected. The car
From playlist Graph Theory
9 3 Random Contraction Algorithm 9 min
From playlist Algorithms 1
From playlist 3d graphs
Graphing a linear inequality when it is in slope intercept form
👉 Learn how to graph linear inequalities written in slope-intercept form. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequal
From playlist Graph Linear Inequalities in Two Variables
Graphing a linear inequality using slope intercept form
👉 Learn how to graph linear inequalities written in slope-intercept form. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequal
From playlist Graph Linear Inequalities in Two Variables
What is the difference between an open and closed point for an inequality
👉 Learn how to graph linear inequalities. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequality is 'excluded' (when less tha
From playlist Graph Linear Inequalities in Two Variables
Summary for graphing linear inequalities
👉 Learn how to graph linear inequalities. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequality is 'excluded' (when less tha
From playlist Graph Linear Inequalities in Two Variables
Lecture 7. Graph partitioning algorithms.
Network Science 2021 @ HSE http://www.leonidzhukov.net/hse/2021/networks/
From playlist Network Science, 2021
Approximating Max Cut with Subexponential Linear Programs - Tselil Schramm
Computer Science/Discrete Mathematics Seminar I Topic: Approximating Max Cut with Subexponential Linear Programs Speaker: Tselil Schramm Affiliation: Stanford University Date: March 29, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Introduction to SNA. Lecture 5. Network communities.
Cohesive subgroups. Graph cliques. Network communities. Graph partitioning. Modularity. Edge Betweenness. Spectral partitioning. Modularity maximization. Heuristic methods. Label propagation. Fast community unfolding. Walktrap. Lecture slides: http://www.leonidzhukov.net/hse/2015/sna/lect
From playlist Introduction to SNA
Network Science. Lecture11.Graph partitioning algorithms
Graph partitioning algorithms Lecture slides: http://www.leonidzhukov.net/hse/2020/networks/lectures/lecture11.pdf
From playlist Network Science. Module 2, 2020
Tina Eliassi-Rad - The Pitfalls of Using ML-based Optimization - IPAM at UCLA
Recorded 03 March 2023. Tina Eliassi-Rad of Northeastern University, Computer Science & Network Science presents "The Pitfalls of Using ML-based Optimization" at IPAM's Artificial Intelligence and Discrete Optimization Workshop. Abstract: I will describe two graph problems where ML-based o
From playlist 2023 Artificial Intelligence and Discrete Optimization
Alexandra Kolla - Quantum Approximate Optimization Algorithm (QAOA) and Local Max-Cut - IPAM at UCLA
Recorded 27 January 2022. Alexandra Kolla of the University of California, Santa Cruz, presents "Quantum Approximate Optimization Algorithm (QAOA) and Local Max-Cut" at IPAM's Quantum Numerical Linear Algebra Workshop. Abstract: We will discuss methods to determine how good of an approxima
From playlist Quantum Numerical Linear Algebra - Jan. 24 - 27, 2022
Rico Zenklusen: An O(1)-approximation for minimum spanning tree interdiction
Rico Zenklusen: An O(1)-approximation for minimum spanning tree interdiction Network interdiction studies the maximum impact that a removal of a limited number of edges or vertices can have on a graph optimization problem. Most interdiction problems are NP-hard, and only little is known a
From playlist HIM Lectures 2015
Jerome Darbon - Algorithms for Non-Local Filtering; application CryoElectron & biological microscopy
Recorded 15 September 2022. Jerome Darbon of Brown University presents "Efficient algorithms for Non-Local Filtering and applications to Cryo-Electron microscopy and biological microscopy" at IPAM's Computational Microscopy Tutorials. Abstract: We present fast and scalable algorithms for n
From playlist Tutorials: Computational Microscopy 2022
Graphical models in machine learning, networks and quantification – A. Bertozzi – ICM2018
Mathematics in Science and Technology Invited Lecture 17.10 Graphical models in machine learning, networks and uncertainty quantification Andrea Bertozzi Abstract: This paper is a review article on semi-supervised and unsupervised graph models for classification using similarity graphs a
From playlist Mathematics in Science and Technology
Graphing an inequality when the slope is one ex 6
👉 Learn how to graph linear inequalities written in slope-intercept form. Linear inequalities are graphed the same way as linear equations, the only difference being that one side of the line that satisfies the inequality is shaded. Also broken line (dashes) is used when the linear inequal
From playlist Graph Linear Inequalities in Two Variables