Graph Algorithms
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture 10: April 20, 2005 Perfect Graphs
Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect. -
I?'!'-Comparability Graphs
Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors). -
Math 7410 Graph Theory
Math 7410 Graph Theory Bogdan Oporowski Department of Mathematics Louisiana State University April 14, 2021 Definition of a graph Definition 1.1 A graph G is a triple (V,E, I) where ◮ V (or V (G)) is a finite set whose elements are called vertices; ◮ E (or E(G)) is a finite set disjoint from V whose elements are called edges; and ◮ I, called the incidence relation, is a subset of V E in which each edge is × in relation with exactly one or two vertices. v2 e1 v1 Example 1.2 ◮ V = v1, v2, v3, v4 e5 { } e2 e4 ◮ E = e1,e2,e3,e4,e5,e6,e7 { } e6 ◮ I = (v1,e1), (v1,e4), (v1,e5), (v1,e6), { v4 (v2,e1), (v2,e2), (v3,e2), (v3,e3), (v3,e5), e7 v3 e3 (v3,e6), (v4,e3), (v4,e4), (v4,e7) } Simple graphs Definition 1.3 ◮ Edges incident with just one vertex are loops. ◮ Edges incident with the same pair of vertices are parallel. ◮ Graphs with no parallel edges and no loops are called simple. v2 e1 v1 e5 e2 e4 e6 v4 e7 v3 e3 Edges of a simple graph can be described as v e1 v 2 1 two-element subsets of the vertex set. Example 1.4 e5 e2 e4 E = v1, v2 , v2, v3 , v3, v4 , e6 {{ } { } { } v1, v4 , v1, v3 . v4 { } { }} v e7 3 e3 Note 1.5 Graph Terminology Definition 1.6 ◮ The graph G is empty if V = , and is trivial if E = . ∅ ∅ ◮ The cardinality of the vertex-set of a graph G is called the order of G and denoted G . -
Bayesian Network - Wikipedia
2019/9/16 Bayesian network - Wikipedia Bayesian network A Bayesian network, Bayes network, belief network, decision network, Bayes(ian) model or probabilistic directed acyclic graphical model is a probabilistic graphical model (a type of statistical model) that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network A simple Bayesian network. Rain could represent the probabilistic relationships between diseases and influences whether the sprinkler symptoms. Given symptoms, the network can be used to compute the is activated, and both rain and probabilities of the presence of various diseases. the sprinkler influence whether the grass is wet. Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. Contents Graphical model Example Inference and learning Inferring unobserved variables Parameter learning Structure learning Statistical introduction Introductory examples Restrictions on priors Definitions and concepts Factorization definition Local Markov property Developing Bayesian networks Markov blanket Causal networks Inference complexity and approximation algorithms Software History See also Notes References https://en.wikipedia.org/wiki/Bayesian_network 1/14 2019/9/16 Bayesian network - Wikipedia Further reading External links Graphical model Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. -
Gestaltungspotenzial Von Digitalen Compositingsystemen
Gestaltungspotenzial von digitalen Compositingsystemen Thorsten Wolf Diplomarbeit Wintersemester 03/04 1. Betreuer Herr Prof. Martin Aichele 2. Betreuer Herr Prof. Christian Fries Fachhochschule Furtwangen Fachbereich Digitale Medien „Das vielleicht größte Missverständnis über die Fotografie kommt in den Worten ‚die Kamera lügt nicht‛ zum Ausdruck. Genau das Gegenteil ist richtig. Die weitaus meisten Fotos sind ‚Lügen‛ in dem Sinne, daß sie nicht vollkommen der Wirklichkeit entsprechen: sie sind zweidimensionale Abbildungen dreidimensionaler Objekte, Schwarzweißbilder farbiger Wirklichkeit, ‚starre‛ Fotos bewegter Objekte. … “ [Kan78] S. 54f Für Mama und Papa, of course. Eidesstattliche Erklärung i Eidesstattliche Erklärung Ich, Thorsten Wolf, erkläre hiermit an Eides statt, dass ich die vorliegende Diplomarbeit selbstständig und ohne unzulässige fremde Hilfe angefertigt habe. Alle verwendeten Quellen und Hilfsmittel sind angegeben. Furtwangen, 24. Februar 2004 Thorsten Wolf Vorwort iii Vorwort In meiner Diplomarbeit „Gestaltungspotenzial von digitalen Compositingsystemen“ untersuche ich den vielseitigen visuellen Bereich der Medieninformatik. In der vorliegenden Arbeit sollen die gestalterischen Potenziale von digitalem Compositing ausgelotet werden. Hierzu untersuche ich theoretisch wie praktisch die digitalen Bildverarbeitungsverfahren und –möglichkeiten für analoge und digitale Bildquellen. Mein besonderes Augenmerk liegt hierbei auf dem Bereich der Bewegtbildgestaltung durch digitale Compositingsysteme. Diese Arbeit entstand in enger Zusammenarbeit mit der Firma on line Video 46 AG, Zürich. Besonderen Dank möchte ich Herrn Richard Rüegg, General Manager, für seine Unterstützung und allen Mitarbeitern, die mir in technischen Fragestellung zur Seite standen, aussprechen. Des weiteren bedanke ich mich bei Patrischa Freuler, Marian Kaiser, Marianne Klein, Jörg Volkmar und Tanja Wolf für ihre Unterstützung während der Diplomarbeitszeit. Für die gute Betreuung möchte ich meinen beiden Tutoren, Herrn Prof. Martin Aichele (Erstbetreuer) und Prof. -
Domination in Circle Graphs
Circles graphs Dominating set Some positive results Open Problems Domination in circle graphs Nicolas Bousquet Daniel Gon¸calves George B. Mertzios Christophe Paul Ignasi Sau St´ephanThomass´e WG 2012 Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems 1 Circles graphs 2 Dominating set 3 Some positive results 4 Open Problems Domination in circle graphs W [1]-hardness Under some algorithmic hypothesis, the W [1]-hard problems do not admit FPT algorithms. Circles graphs Dominating set Some positive results Open Problems Parameterized complexity FPT A problem parameterized by k is FPT (Fixed Parameter Tractable) iff it admits an algorithm which runs in time Poly(n) · f (k) for any instances of size n and of parameter k. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Parameterized complexity FPT A problem parameterized by k is FPT (Fixed Parameter Tractable) iff it admits an algorithm which runs in time Poly(n) · f (k) for any instances of size n and of parameter k. W [1]-hardness Under some algorithmic hypothesis, the W [1]-hard problems do not admit FPT algorithms. Domination in circle graphs Circles graphs Dominating set Some positive results Open Problems Circle graphs Circle graph A circle graph is a graph which can be represented as an intersection graph of chords in a circle. 3 3 4 2 5 4 2 6 1 1 5 7 7 6 Domination in circle graphs Independent dominating sets. Connected dominating sets. Total dominating sets. All these problems are NP-complete. Circles graphs Dominating set Some positive results Open Problems Dominating set 3 4 5 2 6 1 7 Dominating set Set of chords which intersects all the chords of the graph. -
An Introduction to Factor Graphs
An Introduction to Factor Graphs Hans-Andrea Loeliger MLSB 2008, Copenhagen 1 Definition A factor graph represents the factorization of a function of several variables. We use Forney-style factor graphs (Forney, 2001). Example: f(x1, x2, x3, x4, x5) = fA(x1, x2, x3) · fB(x3, x4, x5) · fC(x4). fA fB x1 x3 x5 x2 x4 fC Rules: • A node for every factor. • An edge or half-edge for every variable. • Node g is connected to edge x iff variable x appears in factor g. (What if some variable appears in more than 2 factors?) 2 Example: Markov Chain pXYZ(x, y, z) = pX(x) pY |X(y|x) pZ|Y (z|y). X Y Z pX pY |X pZ|Y We will often use capital letters for the variables. (Why?) Further examples will come later. 3 Message Passing Algorithms operate by passing messages along the edges of a factor graph: - - - 6 6 ? ? - - - - - ... 6 6 ? ? 6 6 ? ? 4 A main point of factor graphs (and similar graphical notations): A Unified View of Historically Different Things Statistical physics: - Markov random fields (Ising 1925) Signal processing: - linear state-space models and Kalman filtering: Kalman 1960. - recursive least-squares adaptive filters - Hidden Markov models: Baum et al. 1966. - unification: Levy et al. 1996. Error correcting codes: - Low-density parity check codes: Gallager 1962; Tanner 1981; MacKay 1996; Luby et al. 1998. - Convolutional codes and Viterbi decoding: Forney 1973. - Turbo codes: Berrou et al. 1993. Machine learning, statistics: - Bayesian networks: Pearl 1988; Shachter 1988; Lauritzen and Spiegelhalter 1988; Shafer and Shenoy 1990. 5 Other Notation Systems for Graphical Models Example: p(u, w, x, y, z) = p(u)p(w)p(x|u, w)p(y|x)p(z|x). -
Indian Journal of Science ANALYSIS International Journal for Science ISSN 2319 – 7730 EISSN 2319 – 7749
Indian Journal of Science ANALYSIS International Journal for Science ISSN 2319 – 7730 EISSN 2319 – 7749 Graph colouring problem applied in genetic algorithm Malathi R Assistant Professor, Dept of Mathematics, Scsvmv University, Enathur, Kanchipuram, Tamil Nadu 631561, India, Email Id: [email protected] Publication History Received: 06 January 2015 Accepted: 05 February 2015 Published: 18 February 2015 Citation Malathi R. Graph colouring problem applied in genetic algorithm. Indian Journal of Science, 2015, 13(38), 37-41 ABSTRACT In this paper we present a hybrid technique that applies a genetic algorithm followed by wisdom of artificial crowds that approach to solve the graph-coloring problem. The genetic algorithm described here, utilizes more than one parent selection and mutation methods depending u p on the state of fitness of its best solution. This results in shifting the solution to the global optimum, more quickly than using a single parent selection or mutation method. The algorithm is tested against the standard DIMACS benchmark tests while, limiting the number of usable colors to the chromatic numbers. The proposed algorithm succeeded to solve the sample data set and even outperformed a recent approach in terms of the minimum number of colors needed to color some of the graphs. The Graph Coloring Problem (GCP) is also known as complete problem. Graph coloring also includes vertex coloring and edge coloring. However, mostly the term graph coloring refers to vertex coloring rather than edge coloring. Given a number of vertices, which form a connected graph, the objective is that to color each vertex so that if two vertices are connected in the graph (i.e. -
Induced Minors and Well-Quasi-Ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck
Induced minors and well-quasi-ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck To cite this version: Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck. Induced minors and well-quasi-ordering. EuroComb: European Conference on Combinatorics, Graph Theory and Appli- cations, Aug 2015, Bergen, Norway. pp.197-201, 10.1016/j.endm.2015.06.029. lirmm-01349277 HAL Id: lirmm-01349277 https://hal-lirmm.ccsd.cnrs.fr/lirmm-01349277 Submitted on 27 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Available online at www.sciencedirect.com www.elsevier.com/locate/endm Induced minors and well-quasi-ordering Jaroslaw Blasiok 1,2 School of Engineering and Applied Sciences, Harvard University, United States Marcin Kami´nski 2 Institute of Computer Science, University of Warsaw, Poland Jean-Florent Raymond 2 Institute of Computer Science, University of Warsaw, Poland, and LIRMM, University of Montpellier, France Th´eophile Trunck 2 LIP, ENS´ de Lyon, France Abstract AgraphH is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor- free. -
Arxiv:1907.08517V3 [Math.CO] 12 Mar 2021
RANDOM COGRAPHS: BROWNIAN GRAPHON LIMIT AND ASYMPTOTIC DEGREE DISTRIBUTION FRÉDÉRIQUE BASSINO, MATHILDE BOUVEL, VALENTIN FÉRAY, LUCAS GERIN, MICKAËL MAAZOUN, AND ADELINE PIERROT ABSTRACT. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0; 1]. We finally an- alyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combi- natorial arguments, including the symbolic method and singularity analysis. 1. INTRODUCTION 1.1. Motivation. Random graphs are arguably the most studied objects at the interface of com- binatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, e.g. perfect graphs [MY19], planar graphs [Noy14], graphs embeddable in a surface of given genus [DKS17], graphs in subcritical classes [PSW16], hereditary classes [HJS18] or addable classes [MSW06, CP19]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings). Cographs were introduced in the seventies by several authors independently, see e.g. -
VFX Prime 2018-19 Course Code: OV-3103 Course Category : Career VFX INDUSTRY
Product Note: VFX Prime 2018-19 Course Code: OV-3103 Course Category : Career VFX INDUSTRY Indian VFX Industry grew from INR 2,320 Crore in 2016 to reach INR 3,130 Crore in 2017.The Industry is expected to grow nearly double to INR 6,350 Crore by 2020. Where reality meets and blends with the imaginary, it is there that VFX begins. The demand for VFX has been rising relentlessly with the production of movies and television shows set in fantasy worlds with imaginary creatures like dragons, magical realms, extra-terrestrial planets and galaxies, and more. VFX can transform the ordinary into something extraordinary. Have you ever been fascinated by films like Transformers, Dead pool, Captain America, Spiderman, etc.? Then you must know that a number of Visual Effects are used in these films. Now the VFX industry is on the verge of changing with the introduction of new tools, new concepts, and ideas. Source:* FICCI-EY Media & Entertainment Report 2018 INDUSTRY TRENDS VFX For Television Episodic Series SONY Television's Show PORUS showcases state-of-the-art Visual Effects to be seen on Television. Based on the tale of King Porus, who fought against Alexander, The Great to stop him from invading India, the show is said to have been made on a budget of Rs500 crore. VFX-based Content for Digital Platforms like Amazon & Netflix Popular web series like House of Cards, Game of Thrones, Suits, etc. on streaming platforms such as Netflix, Amazon Prime, Hot star and many more are unlike any conventional television series. They are edgy and fresh, with high production values, State-of-the-art Visual Effects, which are only matched with films, and are now a rage all over the world. -
Bidimensionality Theory and Algorithmic Graph Minor Theory Lecture Notes for Mohammadtaghi Hajiaghayi’S Tutorial
Bidimensionality Theory and Algorithmic Graph Minor Theory Lecture Notes for MohammadTaghi Hajiaghayi’s Tutorial Mareike Massow,∗ Jens Schmidt,† Daria Schymura,‡ Siamak Tazari§ MDS Fall School Blankensee October 2007 1 Introduction Dealing with Hard Graph Problems Many graph problems cannot be computed in polynomial time unless P = NP , which most computer scientists and mathematicians doubt. Examples are the Traveling Salesman Problem (TSP), vertex cover, and dominating set. TSP is to find a Hamiltonian cycle with the least weight in a complete weighted graph. A vertex cover of a graph is a subset of the vertices that covers all edges. A dominating set in a graph is a subset of the vertices such that each vertex is contained in the set or has a neighbour in the set. The decision problem is to answer the question whether there is a vertex cover resp. a dominating set with at most k elements. The optimization problem is to find a vertex cover resp. a dominating set of minimal size. How can we solve these problems despite their computational complexity? The four main theoretical approaches to handle NP -hard problems are the following. • Average case: Prove that an algorithm is efficient in the expected case. • Special instances: A problem is solved efficiently for a special graph class. For example, graph isomor- phism is computable in linear time on planar graphs whereas graph isomorphism in general is known to be in NP and suspected not to be in P . • Approximation algorithms: Approximate the optimal solution within a constant factor c, or even within 1 + . An algorithmic scheme that gives for each > 0 a polynomial-time (1 + )-approximation is called a polynomial-time approximation scheme (PTAS).