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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. -
Matchings in Balanced Hypergraphs
Matchings in balanced hypergraphs Von der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker Robert Berthold Scheidweiler aus Duren-Birkesdorf.¨ Berichter: Universitatsprofessor¨ Dr. Eberhard Triesch Universitatsprofessor¨ Dr. Arie M.C.A. Koster Tag der mundlichen¨ Prufung:¨ 19. April 2011 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Danksagung In den letzten funf¨ Jahren habe ich am Lehrstuhl II fur¨ Mathematik der RWTH Aachen Uni- versity die vorliegende Dissertation verfasst. Einigen gebuhrt¨ fur¨ ihre Unterstutzung¨ und Hilfe wahrend¨ dieser Zeit besonderer Dank. Zuallererst mochte¨ ich mich bei dem Betreuer meiner Dissertation und meinem Chef, Eberhard Triesch, bedanken. Durch ihn habe ich das Thema dieser Arbeit erhalten, das mir sehr ans Herz gewachsen ist. Er hat mir bei meinen Forschungen immer mit Rat und Tat zur Seite gestanden und mich auch bei langer¨ andauern- den Durststrecken niemals unter Druck gesetzt. Seine positive Unterstutzung¨ und geduldige Hilfe haben mich motiviert, diese Arbeit zu vollenden. Weiterhin mochte¨ ich mich bei Arie Koster, meinem Zweitgutachter, bedanken. Mehrfach hat er im Verlauf meiner Promotion Anregungen gegeben, die dann in die Dissertation eingeflossen sind. Vor der endgultigen¨ Ab- gabe hat er durch seine Verbesserungsvorschlage,¨ fur¨ die ich sehr dankbar bin, zur jetzigen Form der Arbeit beigetragen. Danken mochte¨ ich außerdem Bert Randerath, der mir half, einige Startschwierigkeiten zu uberwinden,¨ als ich begann, die balancierten Hypergraphen zu erforschen. Hartmut Fuhr¨ hat sehr viel Zeit darauf verwendet, mir die harmonische Analysis naher¨ zu bringen. Seine Bemuhungen¨ haben meine Promotion weiter voran gebracht. -
COLORING and DEGENERACY for DETERMINING VERY LARGE and SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Bachelor of Science, Chittag
COLORING AND DEGENERACY FOR DETERMINING VERY LARGE AND SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Bachelor of Science, Chittagong University of Engineering & Technology, 2013 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Ashraful Huq Suny, 2016 COLORING AND DEGENERACY FOR DETERMINING VERY LARGE AND SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Date of Defence: December 19, 2016 Dr. Shahadat Hossain Supervisor Professor Ph.D. Dr. Daya Gaur Committee Member Professor Ph.D. Dr. Robert Benkoczi Committee Member Associate Professor Ph.D. Dr. Howard Cheng Chair, Thesis Examination Com- Associate Professor Ph.D. mittee Dedication To My Parents. iii Abstract Estimation of large sparse Jacobian matrix is a prerequisite for many scientific and engi- neering problems. It is known that determining the nonzero entries of a sparse matrix can be modeled as a graph coloring problem. To find out the optimal partitioning, we have proposed a new algorithm that combines existing exact and heuristic algorithms. We have introduced degeneracy and maximum k-core for sparse matrices to solve the problem in stages. Our combined approach produce better results in terms of partitioning than DSJM and for some test instances, we report optimal partitioning for the first time. iv Acknowledgments I would like to express my deep acknowledgment and profound sense of gratitude to my supervisor Dr. Shahadat Hossain, for his continuous guidance, support, cooperation, and persistent encouragement throughout the journey of my MSc program. -
The Harmonious Coloring Problem Is NP-Complete for Interval and Permutation Graphsଁ Katerina Asdre, Kyriaki Ioannidou, Stavros D
Discrete Applied Mathematics 155 (2007) 2377–2382 www.elsevier.com/locate/dam Note The harmonious coloring problem is NP-complete for interval and permutation graphsଁ Katerina Asdre, Kyriaki Ioannidou, Stavros D. Nikolopoulos Department of Computer Science, University of Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece Received 4 November 2006; received in revised form 25 May 2007; accepted 6 July 2007 Available online 14 September 2007 Abstract In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs. © 2007 Elsevier B.V. All rights reserved. Keywords: Harmonious coloring; Harmonious chromatic number; Achromatic number; Interval graphs; Permutation graphs; NP-completeness 1. Introduction Many NP-complete problems on arbitrary graphs admit polynomial time algorithms when restricted to the classes of interval graphs and cographs; NP-complete problems for these two classes of graphs that become solvable in polynomial time can be found in [1,3,7,12,15,16]. However, the pair-complete coloring problem, which is NP-hard on arbitrary graphs [17], remains NP-complete when restricted to graphs that are simultaneously interval and cographs [4].Apair- complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge, while the achromatic number (G) is the largest integer k for which G admits a pair-complete coloring with k colors. -
3-Coloring Gnp3 in Polynomial Expected Time
Mathematisch-Naturwissenschaftliche Fakultat¨ II Institut fur¨ Informatik Lehrstuhl fur¨ Algorithmen und Komplexit¨at Coloring sparse random k-colorable graphs in polynomial expected time Diplomarbeit eingereicht von: Julia B¨ottcher Gutachter: Dr. Amin Coja-Oghlan Dr. Mihyun Kang Berlin, den 5. Januar 2005 Erkl¨arung Hiermit erkl¨are ich, die vorliegende Arbeit selbstst¨andig verfasst und keine anderen als die angegebenen Hilfsmittel verwendet zu haben. Ich bin damit einverstanden, dass die vorliegen- de Arbeit in der Zentralbibliothek Naturwissenschaften der Humboldt-Universit¨at zu Berlin ¨offentlich ausgelegt wird. Berlin, den 5. Januar 2005 Julia B¨ottcher Zusammenfassung Nach wie vor stellt das Graphenf¨arbungsproblem eine der anspruchsvollsten algorithmischen Aufgaben innerhalb der Graphentheorie dar. So blieb bisher nicht nur die Suche nach effizienten Methoden, die dieses Problem exakt l¨osen, erfolglos. Gem¨aß eines Resultates von Feige und Kilian [22] ist auch die Existenz von signifikant besseren Approximationsalgorithmen als dem Trivialen, der jedem Knoten eine eigene Farbe zuordnet, sehr unwahrscheinlich. Diese Tatsache motiviert die Suche nach Algorithmen, die das Problem entweder nicht in allen sondern nur den meisten F¨allen entscheiden, dafur¨ aber von polynomieller Laufzeitkomplexit¨at sind, oder aber stets eine korrekte L¨osung ausgeben, eine polynomielle Laufzeit jedoch nur im Durchschnitt garantieren. Ein Algorithmus der ersterem Paradigma folgt, wurde von Alon und Kahale [2] fur¨ die fol- gende Klasse zuf¨alliger k-f¨arbbarer Graphen entwickelt: Konstruiere einen Graphen mit Gn,p,k einer Knotenmenge V der Kardinalit¨at n durch Partitionierung von V in k Mengen gleicher Gr¨oße und anschließendes Hinzufugen¨ der m¨oglichen Kanten zwischen diesen Mengen mit Wahrscheinlichkeit jeweils p. -
Oriented Hypergraphs I: Introduction and Balance Arxiv:1210.0943V1
Oriented Hypergraphs I: Introduction and Balance Lucas J. Rusnak∗ Department of Mathematics Texas State University San Marcos, Texas, U.S.A. [email protected] Submitted: Sept 30, 2012; Accepted: MMM DD, YYYY; Published: XX Mathematics Subject Classifications: 05C22, 05C65, 05C75 Abstract An oriented hypergraph is an oriented incidence structure that extends the con- cept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families { balanced, bal- anceable, and unbalanceable { and we obtain a complete classification of the bal- anced circuits of oriented hypergraphs. Keywords: Oriented hypergraph; balanced hypergraph; balanced matrix; signed hypergraph 1 Introduction Oriented hypergraphs have recently appeared in [6] as an extension of the signed graphic incidence, adjacency, and Laplacian matrices to examine walk counting. This paper fur- ther expands the theory of oriented hypergraphs by examining the extension of the cycle arXiv:1210.0943v1 [math.CO] 2 Oct 2012 space of a graph to oriented hypergraphs, and we obtain a classification of the balanced minimally dependent columns of the incidence matrix of an oriented hypergraph. It is known that the cycle space of a graph characterizes the dependencies of the graphic matroid and the minimal dependencies, or circuits, are the edge sets of the simple cycles of the graph. Oriented hypergraphs have a natural division into three categories: balanced, balanceable, and unbalanceable. The family of balanced oriented hypergraphs contain graphs, so a characterization of the balanced circuits of oriented hypergraphs can be regarded as an extension of the following theorem: ∗A special thanks to Thomas Zaslavsky and Gerard Cornu´ejolsfor their feedback. -
Graph Theory Graph Theory (II)
J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. -
High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality
High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality Maciej Besta1, Armon Carigiet1, Zur Vonarburg-Shmaria1, Kacper Janda2, Lukas Gianinazzi1, Torsten Hoefler1 1Department of Computer Science, ETH Zurich 2Faculty of Computer Science, Electronics and Telecommunications, AGH-UST Abstract—We develop the first parallel graph coloring heuris- to their degrees, random (R) [26] which chooses vertices tics with strong theoretical guarantees on work and depth and uniformly at random, incidence-degree (ID) [1] which picks coloring quality. The key idea is to design a relaxation of the vertices with the largest number of uncolored neighbors first, vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This saturation-degree (SD) [27], where a vertex whose neighbors introduces a tunable amount of parallelism into the degeneracy use the largest number of distinct colors is chosen first, and ordering that is otherwise hard to parallelize. This simple idea smallest-degree-last (SL) [28] that removes lowest degree enables significant benefits in several key aspects of graph color- vertices, recursively colors the resulting graph, and then colors ing. For example, one of our algorithms ensures polylogarithmic the removed vertices. All these ordering heuristics, combined depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining work- with Greedy, have the inherent problem of no parallelism. efficiency. In addition to provable guarantees, the developed Jones and Plassmann combined this line of work with earlier algorithms have competitive run-times for several real-world parallel schemes for deriving maximum independent sets [29], graphs, while almost always providing superior coloring quality. -
The Second Malta Conference in Graph Theory and Combinatorics 2017
The Second Malta Conference in Graph Theory and Combinatorics 2017 2MCGTC-2017 in honour of the 75th birthday of Professor Stanley Fiorini 26{30 June 2017 ii Welcome Address Mer~ba! We are honoured that you chose to join us for The Second Malta Confer- ence in Graph Theory and Combinatorics. This conference is commemorating the 75th birthday of Professor Stanley Fiorini, who introduced graph theory and combinatorics at the University of Malta. Many of you may be asking when the previous Malta Conference was held? The First Malta Conference on Graphs and Combinatorics was held during the period 28 May { 2 June, 1990, at the Suncrest Hotel, also in Qawra, St Paul's Bay. It differed from a number of similar conferences held in the central Mediterranean region at that time in that it consisted of three types of lectures. L´aszl´oLov´aszand Carsten Thomassen delivered two instructional courses of five one-hour lectures each; the former was A survey of independent sets in graphs and the latter was on Embeddings of graphs. There were also four invited speakers, namely L.W. Beineke, N.L. Biggs, R. Graham and D.J.A. Welsh, each of whom gave a one-hour lecture. The third type of talks were the 20-minute contributed talks running in two parallel sessions and given by 39 speakers. Volume 124 (1994) of the journal Discrete Mathematics was a special edition dedicated to this conference; it was edited by Stanley Fiorini and Josef Lauri, and it consisted of 22 selected papers. Twenty-seven years later we are gathered here for the second such conference organ- ised in the Island of Malta. -
A Characterization of Oriented Hypergraphic Balance Via Signed Weak Walks
Linear Algebra and its Applications 485 (2015) 442–453 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa A characterization of oriented hypergraphic balance via signed weak walks Vinciane Chen a, Angeline Rao a, Lucas J. Rusnak b,∗, Alex Yang a a Texas State Mathworks, Texas State University, San Marcos, TX 78666, USA b Department of Mathematics, Texas State University, San Marcos, TX 78666, USA a r t i c l e i n f o a b s t r a c t Article history: An oriented hypergraph is a hypergraph where each vertex- Received 23 August 2014 edge incidence is given a label of +1 or −1, and each adjacency Accepted 1 August 2015 is signed the negative of the product of the incidences. An ori- Available online xxxx ented hypergraph is balanced if the product of the adjacencies Submitted by R. Brualdi in each circle is positive. MSC: We provide a combinatorial interpretation for entries of kth 05C50 power of the oriented hypergraphic Laplacian via the number 05C65 of signed weak walks of length k. Using closed weak walks we 05C22 prove a new characterization of balance for oriented hyper- graphs and matrices that generalizes Harary’s Theorem for Keywords: signed graphs. Oriented hypergraph © 2015 Published by Elsevier Inc. Laplacian matrix Balanced hypergraphs 1. Introduction An oriented hypergraph is a signed incidence structure where each vertex-edge inci- dence is given a label of +1 or −1, and each adjacency is signed the negative of their incidence product. A signed graph is an oriented hypergraph where each edge is con- * Corresponding author. -
The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs
Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany ISABEL BECKENBACH,MEIKE HATZEL,SEBASTIAN WIEDERRECHT The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs ZIB Report 18-61 (December 2018) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs Isabel Beckenbach (Zuse Institute Berlin) [email protected] Meike Hatzel (TU Berlin) [email protected] Sebastian Wiederrecht (TU Berlin) [email protected] The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on the tight cut decomposition is that the list of bricks obtained is the same independent of the choice of tight cuts made during the tight cut decomposition procedure. This implies that finding a tight cut decomposition is polynomial time equivalent to finding a single tight cut. We generalise the notions of a tight cut, a tight cut contraction and a tight cut decomposition to hypergraphs. -
Jgrapht--A Java Library for Graph Data Structures and Algorithms
JGraphT - A Java library for graph data structures and algorithms Dimitrios Michail1, Joris Kinable2,3, Barak Naveh4, and John V Sichi5 1Dept. of Informatics and Telematics Harokopio University of Athens, Greece [email protected] 2Dept. of Industrial Engineering and Innovation Sciences Eindhoven University of Technology, The Netherlands [email protected] 3Robotics Institute Carnegie Mellon University, Pittsburgh, USA 4barak [email protected] 5The JGraphT project [email protected] Abstract Mathematical software and graph-theoretical algorithmic packages to efficiently model, analyze and query graphs are crucial in an era where large-scale spatial, societal and economic network data are abundantly available. One such package is JGraphT, a pro- gramming library which contains very efficient and generic graph data-structures along with a large collection of state-of-the-art algorithms. The library is written in Java with stability, interoperability and performance in mind. A distinctive feature of this library is its ability to model vertices and edges as arbitrary objects, thereby permitting natural representations of many common networks including transportation, social and biologi- cal networks. Besides classic graph algorithms such as shortest-paths and spanning-tree algorithms, the library contains numerous advanced algorithms: graph and subgraph iso- morphism; matching and flow problems; approximation algorithms for NP-hard problems arXiv:1904.08355v2 [cs.DS] 3 Feb 2020 such as independent set and TSP; and several more exotic algorithms such as Berge graph detection. Due to its versatility and generic design, JGraphT is currently used in large- scale commercial products, as well as non-commercial and academic research projects. In this work we describe in detail the design and underlying structure of the library, and discuss its most important features and algorithms.