Antimatroids and Balanced Pairs
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Knowledge Spaces Applications in Education Knowledge Spaces
Jean-Claude Falmagne · Dietrich Albert Christopher Doble · David Eppstein Xiangen Hu Editors Knowledge Spaces Applications in Education Knowledge Spaces Jean-Claude Falmagne • Dietrich Albert Christopher Doble • David Eppstein • Xiangen Hu Editors Knowledge Spaces Applications in Education Editors Jean-Claude Falmagne Dietrich Albert School of Social Sciences, Department of Psychology Dept. Cognitive Sciences University of Graz University of California, Irvine Graz, Austria Irvine, CA, USA Christopher Doble David Eppstein ALEKS Corporation Donald Bren School of Information Irvine, CA, USA & Computer Sciences University of California, Irvine Irvine, CA, USA Xiangen Hu Department of Psychology University of Memphis Memphis, TN, USA ISBN 978-3-642-35328-4 ISBN 978-3-642-35329-1 (eBook) DOI 10.1007/978-3-642-35329-1 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013942001 © Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. -
A Forbidden Subgraph Characterization of Some Graph Classes Using Betweenness Axioms
A forbidden subgraph characterization of some graph classes using betweenness axioms Manoj Changat Department of Futures Studies University of Kerala Trivandrum - 695034, India e-mail: [email protected] Anandavally K Lakshmikuttyamma ∗ Department of Futures Studies University of Kerala, Trivandrum-695034, India e-mail: [email protected] Joseph Mathews Iztok Peterin C-GRAF, Department of Futures Studies University of Maribor University of Kerala, Trivandrum-695034, India FEECS, Smetanova 17, e-mail:[email protected] 2000 Maribor, Slovenia [email protected] Prasanth G. Narasimha-Shenoi Geetha Seethakuttyamma Department of Mathematics Department of Mathematics Government College, Chittur College of Engineering Palakkad - 678104, India Karungapally e-mail: [email protected] e-mail:[email protected] Simon Špacapan University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia. e-mail: [email protected]. July 27, 2012 ∗ The work of this author (On leave from Mar Ivanios College, Trivandrum) is supported by the University Grants Commission (UGC), Govt. of India under the FIP scheme. 1 Abstract Let IG(x,y) and JG(x,y) be the geodesic and induced path interval between x and y in a connected graph G, respectively. The following three betweenness axioms are considered for a set V and R : V × V → 2V (i) x ∈ R(u,y),y ∈ R(x,v),x 6= y, |R(u,v)| > 2 ⇒ x ∈ R(u,v) (ii) x ∈ R(u,v) ⇒ R(u,x) ∩ R(x,v)= {x} (iii)x ∈ R(u,y),y ∈ R(x,v),x 6= y, ⇒ x ∈ R(u,v). We characterize the class of graphs for which IG satisfies (i), and the class for which JG satisfies (ii) and the class for which IG or JG satisfies (iii). -
Drawing Graphs and Maps with Curves
Report from Dagstuhl Seminar 13151 Drawing Graphs and Maps with Curves Edited by Stephen Kobourov1, Martin Nöllenburg2, and Monique Teillaud3 1 University of Arizona – Tucson, US, [email protected] 2 KIT – Karlsruhe Institute of Technology, DE, [email protected] 3 INRIA Sophia Antipolis – Méditerranée, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 13151 “Drawing Graphs and Maps with Curves”. The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants. Seminar 07.–12. April, 2013 – www.dagstuhl.de/13151 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases graph drawing, information visualization, computational cartography, computational geometry Digital Object Identifier 10.4230/DagRep.3.4.34 Edited in cooperation with Benjamin Niedermann 1 Executive Summary Stephen Kobourov Martin Nöllenburg Monique Teillaud License Creative Commons BY 3.0 Unported license © Stephen Kobourov, Martin Nöllenburg, and Monique Teillaud Graphs and networks, maps and schematic map representations are frequently used in many fields of science, humanities and the arts. -
(Eds.) Innovations in Bayesian Networks
Dawn E. Holmes and Lakhmi C. Jain (Eds.) InnovationsinBayesianNetworks Studies in Computational Intelligence,Volume 156 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: Vol. 145. Mikhail Ju. Moshkov, Marcin Piliszczuk springer.com and Beata Zielosko Partial Covers, Reducts and Decision Rules in Rough Sets, 2008 Vol. 134. Ngoc Thanh Nguyen and Radoslaw Katarzyniak (Eds.) ISBN 978-3-540-69027-6 New Challenges in Applied Intelligence Technologies, 2008 Vol. 146. Fatos Xhafa and Ajith Abraham (Eds.) ISBN 978-3-540-79354-0 Metaheuristics for Scheduling in Distributed Computing Vol. 135. Hsinchun Chen and Christopher C.Yang (Eds.) Environments, 2008 Intelligence and Security Informatics, 2008 ISBN 978-3-540-69260-7 ISBN 978-3-540-69207-2 Vol. 147. Oliver Kramer Self-Adaptive Heuristics for Evolutionary Computation, 2008 Vol. 136. Carlos Cotta, Marc Sevaux ISBN 978-3-540-69280-5 and Kenneth S¨orensen (Eds.) Adaptive and Multilevel Metaheuristics, 2008 Vol. 148. Philipp Limbourg ISBN 978-3-540-79437-0 Dependability Modelling under Uncertainty, 2008 ISBN 978-3-540-69286-7 Vol. 137. Lakhmi C. Jain, Mika Sato-Ilic, Maria Virvou, George A. Tsihrintzis,Valentina Emilia Balas Vol. 149. Roger Lee (Ed.) and Canicious Abeynayake (Eds.) Software Engineering, Artificial Intelligence, Networking and Computational Intelligence Paradigms, 2008 Parallel/Distributed Computing, 2008 ISBN 978-3-540-79473-8 ISBN 978-3-540-70559-8 Vol. 138. Bruno Apolloni,Witold Pedrycz, Simone Bassis Vol. 150. Roger Lee (Ed.) and Dario Malchiodi Software Engineering Research, Management and The Puzzle of Granular Computing, 2008 Applications, 2008 ISBN 978-3-540-79863-7 ISBN 978-3-540-70774-5 Vol. -
On Computing Longest Paths in Small Graph Classes
On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not. -
Vertex Deletion Problems on Chordal Graphs∗†
Vertex Deletion Problems on Chordal Graphs∗† Yixin Cao1, Yuping Ke2, Yota Otachi3, and Jie You4 1 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 2 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 3 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, Japan [email protected] 4 School of Information Science and Engineering, Central South University and Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yan- nakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity, G.2.2 Graph Theory Keywords and phrases vertex deletion problem, maximum subgraph, chordal graph, (unit) in- terval graph, split graph, hereditary property, NP-complete, polynomial-time algorithm Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2017.22 1 Introduction Generally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices. -
Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
Discrete Mathematics 310 (2010) 897–910 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Rooted directed path graphs are leaf powers Andreas Brandstädt a, Christian Hundt a, Federico Mancini b, Peter Wagner a a Institut für Informatik, Universität Rostock, D-18051, Germany b Department of Informatics, University of Bergen, N-5020 Bergen, Norway article info a b s t r a c t Article history: Leaf powers are a graph class which has been introduced to model the problem of Received 11 March 2009 reconstructing phylogenetic trees. A graph G D .V ; E/ is called k-leaf power if it admits a Received in revised form 2 September 2009 k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance Accepted 13 October 2009 between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a Available online 30 October 2009 k-leaf power for some k 2 N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a Keywords: given graph is a k-leaf power. Leaf powers Leaf roots We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a Graph class inclusions well-known graph class with absolutely different motivations. After this, we provide the Strongly chordal graphs largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path Fixed tolerance NeST graphs graphs. -
2/3 Conjecture
PROGRESS ON THE 1=3 − 2=3 CONJECTURE By Emily Jean Olson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics | Doctor of Philosophy 2017 ABSTRACT PROGRESS ON THE 1=3 − 2=3 CONJECTURE By Emily Jean Olson Let (P; ≤) be a finite partially ordered set, also called a poset, and let n denote the cardinality of P . Fix a natural labeling on P so that the elements of P correspond to [n] = f1; 2; : : : ; ng. A linear extension is an order-preserving total order x1 ≺ x2 ≺ · · · ≺ xn on the elements of P , and more compactly, we can view this as the permutation x1x2 ··· xn in one-line notation. For distinct elements x; y 2 P , we define P(x ≺ y) to be the proportion 1 of linear extensions of P in which x comes before y. For 0 ≤ α ≤ 2, we say (x; y) is an α-balanced pair if α ≤ P(x ≺ y) ≤ 1 − α: The 1=3 − 2=3 Conjecture states that every finite partially ordered set that is not a chain has a 1=3-balanced pair. This dissertation focuses on showing the conjecture is true for certain types of partially ordered sets. We begin by discussing a special case, namely when a partial order is 1=2-balanced. For example, this happens when the poset has an automorphism with a cycle of length 2. We spend the remainder of the text proving the conjecture is true for some lattices, including Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. -
Introducing Subclasses of Basic Chordal Graphs
Introducing subclasses of basic chordal graphs Pablo De Caria 1 Marisa Gutierrez 2 CONICET/ Universidad Nacional de La Plata, Argentina Abstract Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs. Keywords: Chordal graph, dually chordal graph, clique tree, compatible tree. 1 Introduction 1.1 Definitions For a graph G, V (G) is the set of its vertices and E(G) is the set of its edges. A clique is a maximal set of pairwise adjacent vertices. The family of cliques of G is denoted by C(G). For v ∈ V (G), Cv will denote the family of cliques containing v. All graphs considered will be assumed to be connected. A set S ⊆ V (G) is a uv-separator if vertices u and v are in different connected components of G−S. It is minimal if no proper subset of S has the 1 Email: [email protected] aaURL: www.mate.unlp.edu.ar/~pdecaria 2 Email: [email protected] same property. S is a minimal vertex separator if there exist two nonadjacent vertices u and v such that S is a minimal uv-separator. S(G) will denote the family of all minimal vertex separators of G. -
Graph Layout for Applications in Compiler Construction
Theoretical Computer Science Theoretical Computer Science 2 I7 ( 1999) 175-2 I4 Graph layout for applications in compiler construction G. Sander * Abstract We address graph visualization from the viewpoint of compiler construction. Most data struc- tures in compilers are large, dense graphs such as annotated control flow graph, syntax trees. dependency graphs. Our main focus is the animation and interactive exploration of these graphs. Fast layout heuristics and powerful browsing methods are needed. We give a survey of layout heuristics for general directed and undirected graphs and present the browsing facilities that help to manage large structured graphs. @ l999-Elsevier Science B.V. All rights reserved Keyrror& Graph layout; Graph drawing; Compiler visualization: Force-directed placement; Hierarchical layout: Compound drawing; Fisheye view 1. Introduction We address graph visualization from the viewpoint of compiler construction. Draw- ings of compiler data structures such as syntax trees, control flow graphs, dependency graphs [63], are used for demonstration, debugging and documentation of compilers. In real-world compiler applications, such drawings cannot be produced manually because the graphs are automatically generated, large, often dense, and seldom planar. Graph layout algorithms help to produce drawings automatically: they calculate positions of nodes and edges of the graph in the plane. Our main focus is the animation and interactive exploration of compiler graphs. Thus. fast layout algorithms are required. Animations show the behaviour of an algorithm by a running sequence of drawings. Thus there is not much time to calculate a layout between two subsequent drawings. In interactive exploration, it is annoying if the user has to wait a long time for a layout. -
Forbidden Configurations in Discrete Geometry
FORBIDDEN CONFIGURATIONS IN DISCRETE GEOMETRY This book surveys the mathematical and computational properties of finite sets of points in the plane, covering recent breakthroughs on important problems in discrete geometry and listing many open prob- lems. It unifies these mathematical and computational views using for- bidden configurations, which are patterns that cannot appear in sets with a given property, and explores the implications of this unified view. Written with minimal prerequisites and featuring plenty of fig- ures, this engaging book will be of interest to undergraduate students and researchers in mathematics and computer science. Most topics are introduced with a related puzzle or brain-teaser. The topics range from abstract issues of collinearity, convexity, and general position to more applied areas including robust statistical estimation and network visualization, with connections to related areas of mathe- matics including number theory, graph theory, and the theory of per- mutation patterns. Pseudocode is included for many algorithms that compute properties of point sets. David Eppstein is Chancellor’s Professor of Computer Science at the University of California, Irvine. He has more than 350 publications on subjects including discrete and computational geometry, graph the- ory, graph algorithms, data structures, robust statistics, social network analysis and visualization, mesh generation, biosequence comparison, exponential algorithms, and recreational mathematics. He has been the moderator for data structures and algorithms -
Counting Linear Extensions in Practice: MCMC Versus Exponential Monte Carlo
The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18) Counting Linear Extensions in Practice: MCMC versus Exponential Monte Carlo Topi Talvitie Kustaa Kangas Teppo Niinimaki¨ Mikko Koivisto Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science University of Helsinki Aalto University Aalto University University of Helsinki topi.talvitie@helsinki.fi juho-kustaa.kangas@aalto.fi teppo.niinimaki@aalto.fi mikko.koivisto@helsinki.fi Abstract diagram (Kangas et al. 2016), or the treewidth of the incom- parability graph (Eiben et al. 2016). Counting the linear extensions of a given partial order is a If an approximation with given probability is sufficient, #P-complete problem that arises in numerous applications. For polynomial-time approximation, several Markov chain the problem admits a polynomial-time solution for all posets Monte Carlo schemes have been proposed; however, little is using Markov chain Monte Carlo (MCMC) methods. The known of their efficiency in practice. This work presents an first fully-polynomial time approximation scheme of this empirical evaluation of the state-of-the-art schemes and in- kind for was based on algorithms for approximating the vol- vestigates a number of ideas to enhance their performance. In umes of convex bodies (Dyer, Frieze, and Kannan 1991). addition, we introduce a novel approximation scheme, adap- The later improvements were based on rapidly mixing tive relaxation Monte Carlo (ARMC), that leverages exact Markov chains in the set of linear extensions (Karzanov exponential-time counting algorithms. We show that approx- and Khachiyan 1991; Bubley and Dyer 1999) combined imate counting is feasible up to a few hundred elements on with Monte Carlo counting schemes (Brightwell and Win- various classes of partial orders, and within this range ARMC kler 1991; Banks et al.