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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. -
1 Introduction and Terminology
A Survey of Solved Problems and Applications on Bandwidth, Edgesum and Pro le of Graphs Yung-Ling Lai National Chiayi Teacher College Chiayi, Taiwan, R.O.C. Kenneth Williams Western Michigan University Abstract This pap er provides a survey of results on the exact bandwidth, edge- sum, and pro le of graphs. A bibliographyof work in these areas is pro- vided. The emphasis is on comp osite graphs. This may be regarded as an up date of the original survey of solved bandwidth problems by Chinn, Chvatalova, Dewdney, and Gibbs[10] in 1982. Also several of the applica- tion areas involving these graph parameters are describ ed. 1 Intro duction and terminology For a graph G, V (G) denotes the set of vertices of G and E (G) denotes the set of edges of G. Let G = (V; E ) be a graph on n vertices. A 1-1 mapping f : V ! f1; 2;:::;ng is called a proper numbering of G. The bandwidth B (G) of aproper f numbering f of G is the number B (G)= maxfjf (u) f (v )j : uv 2 E g; f and the bandwidth B(G) of G is the number B (G)= minfB (G): f is a prop er numb ering of Gg: f A prop er numb ering f is called a bandwidth numbering of G if B (G)=B (G). f For example, Figure 1 shows bandwidth numb erings for the graphs P ;C ;K 4 5 1;4 and K . In general, B (P ) = 1, B (C ) = 2, B (K ) = dn=2e and B (K ) = 2;3 n n 1;n m;n m + dn=2e1form n. -
On Bounding the Bandwidth of Graphs with Symmetry
On bounding the bandwidth of graphs with symmetry E.R. van Dam∗ R. Sotirovy Abstract We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite program- ming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. This fixing results in several smaller subproblems that need to be solved to obtain the new bound. In order to efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we de- velop a heuristic approach based on the well-known reverse Cuthill-McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs un- der consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices. Keywords: bandwidth, minimum cut, semidefinite programming, Hamming graphs, John- son graphs, Kneser graphs 1 Introduction For (undirected) graphs, the bandwidth problem (BP) is the problem of labeling the vertices of a given graph with distinct integers such that the maximum difference between the labels of adjacent vertices is minimal. Determining the bandwidth is NP-hard (see [35]) and it remains NP-hard even if it is restricted to trees with maximum degree three (see [17]) or to caterpillars with hair length three (see [34]). -
Exact and Approximate Bandwidth
Exact and Approximate Bandwidth Marek Cygan and Marcin Pilipczuk Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Poland {cygan,malcin}@mimuw.edu.pl Abstract. In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the BANDWIDTH problem. For graphs with treewidth t we present a O(nO(t)2n) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for anyα>0 there exists a (1 + α)-approximation algorithm running in O(exp(c(t + n/α)logn)) time where c is a universal constant. These re- sults seem interesting since Unger has proved that BANDWIDTH does not belong to AP X even when the input graph is a tree (assuming P = NP). So some- what surprisingly, despite Unger’s result it turns out that not only a subexponen- tial constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r,wepresenta(4r − 1)- approximation algorithm that solves BANDWIDTH for an arbitrary input graph ∗ n in O (2 r ) time and polynomial space.1 Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473n ) time and space algorithm. In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Bj¨orklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O∗(2n) time and space, what matches the best known results. -
Counting Subgraphs Via Homomorphisms∗
Counting Subgraphs via Homomorphisms∗ Omid Aminiy Fedor V. Fominz Saket Saurabhx Abstract We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics includ- ing the recent algorithm of Bj¨orklund,Husfeldt and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamil- tonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color coding based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions and the color coding technique, we obtain the following new results: • The number of optimal bandwidth permutations of a graph on n vertices excluding n+o(n) a fixed graph as a minor can be computedp in time O(2 ); in particular in time O(2nn3) for trees and in time 2n+O( n) for planar graphs. • Counting all maximum planar subgraphs, subgraphs of bounded genus, or more gen- erally subgraphs excluding a fixed graph M as a minor can be done in 2O(n) time. • Counting all subtrees with a given maximum degree (a generalization of counting Hamiltonian paths) of a given graph can be done in time 2O(n). • A generalization of Ryser's formula: Let G be a graph with an independent set of size `. Then the number of perfect matchings in G can be found in time O(2n−`n3). -
Exact and Approximate Bandwidth
Theoretical Computer Science 411 (2010) 3701–3713 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Exact and approximate bandwidthI Marek Cygan ∗, Marcin Pilipczuk Department of Mathematics, Computer Science and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland article info a b s t r a c t Article history: Received 17 October 2009 In this paper we gather several improvements in the field of exact and approximate Received in revised form 7 June 2010 exponential time algorithms for the Bandwidth problem. For graphs with treewidth t we Accepted 20 June 2010 present an O.nO.t/2n/ exact algorithm. Communicated by G. Ausiello Moreover, for any two positive integers k ≥ 2; r ≥ 1, we present a .2kr − 1/- n ∗ .k−1/r Keywords: approximation algorithm that solves Bandwidth for an arbitrary input graph in O .k / ∗ Bandwidth time and polynomial space where by O we denote the standard big O notation but omitting Exponential algorithm polynomial factors. Finally, we improve the currently best known exact algorithm for Exact algorithm arbitrary graphs with an O.4:383n/ time and space algorithm. Approximate algorithm In the algorithms for the small treewidth we develop a technique based on the Fast Graph Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Björklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O∗.2n/ time and space, what matches the best known results. ' 2010 Elsevier B.V. All rights reserved. -
Practical Course on Graph Visualization Introduction
Practical Course on Graph Visualization Introduction SUMMER SEMESTER 2014/2015 Tamara Mchedlidze – Martin Nollenburg¨ VS KIT – University of the State of Baden-Wuerttemberg and Institute of Theoreticalwww.kit.edu Informatics TamaraNational Mchedlidze Laboratory – of Practical the Helmholtz Course Association on Graph Visualization Prof. Dr. Dorothea Wagner Organization Instructors Tamara Mchedlidze [email protected] Office 307 Martin Nollenburg¨ [email protected] Office 319 Class meetings Tuesday 11:30 - 13:00 Room SR301 Detailed plan of meetings on the webpage Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Organization Web-page http://i11www.iti.uni-karlsruhe.de/teaching/sommer2015/graphvis/ Information on the web-page Meeting dates (subject to change) Deadlines and goals Literature and reading material (...) Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Practical Course on Graph Visualization Winter Semester Summer Semester Algorithms for Practical Course on Graph Visualization Graph Visualization Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Practical Course on Graph Visualization Winter Semester Summer Semester Algorithms for Practical Course on Graph Visualization Graph Visualization Practical course is part of the module Theory and Practice of Graph Visualization (IN4INGTP) Module consists of the theory lecture (5 credits) and the practical course (5 credits) Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. Dr. Dorothea Wagner Practical Course on Graph Visualization Winter Semester Summer Semester Algorithms for Practical Course on Graph Visualization Graph Visualization Institute of Theoretical Informatics Tamara Mchedlidze – Practical Course on Graph Visualization Prof. -
A Survey on Graph Drawing Beyond Planarity
A Survey on Graph Drawing Beyond Planarity Walter Didimo1, Giuseppe Liotta1, Fabrizio Montecchiani1 1Dipartimento di Ingegneria, Universit`adegli Studi di Perugia, Italy fwalter.didimo,giuseppe.liotta,[email protected] Abstract Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of non-planar graphs in terms of forbidden crossing configurations. Aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems. 1 Introduction In the mid 1980s, the early pioneers of graph drawing had the intuition that a drawing with too many edge crossings is harder to read than a drawing of the same graph with fewer edge crossings (see, e.g., [35, 36, 72, 190]). This intuition was later confirmed by a series of cognitive experimental studies (see, e.g., [180, 181, 203]). As a result, a large part of the existing literature on graph drawing showcases elegant algorithms and sophisticated data structures under the assumption that the input graph is planar, i.e., it admits a drawing without edge crossings. When the input graph is non-planar, crossing minimization heuristics are used to insert a small number of dummy vertices in correspondence of the edge crossings, so to obtain a planarization of the input graph. A crossing-free drawing of the planarization can be computed by using one of the algorithms for planar graphs and then the crossings are reinserted by removing the dummy vertices. This approach is commonly adopted and works well for graphs of relatively small size, up to a few hundred vertices and edges (see, e.g., [87, 152]). -
Crossing Minimization for 1-Page and 2-Page Drawings of Graphs with Bounded Treewidth
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 22, no. 4, pp. 577{606 (2018) DOI: 10.7155/jgaa.00479 Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth Michael J. Bannister 1 David Eppstein 2 1Pinterest, San Francisco, California, USA 2Dept. of Computer Science, University of California, Irvine, USA Abstract We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed- parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth. Submitted: Reviewed: Revised: Accepted: Final: March 2018 August 2018 September 2018 October 2018 October 2018 Published: December 2018 Article type: Communicated by: Regular paper W.S. Evans This material is based upon work supported by the National Science Foundation under Grants CCF-1228639, CCF-1618301, and CCF-1616248, and by the Office of Naval Research under Grant No. N00014-08-1-1015. A preliminary version of this work appeared in the Proceedings of the 22nd International Symposium on Graph Drawing (GD 2014) [6]. E-mail addresses: [email protected] (Michael J. Bannister) [email protected] (David Eppstein) 578 Bannister and Eppstein Crossing Minimization for 1-page and 2-page.. -
Parameterized Complexity and Approximation Algorithms
c The Author 2005. Published by Oxford University Press on behalf of The British Computer Society. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/comjnl/bxh000 Parameterized complexity and approximation algorithms Daniel´ Marx Institut f¨ur Informatik, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, 10099, Berlin, Germany. Email: [email protected] Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type, and propose directions for future research. 1. INTRODUCTION good as an optimum solution. However, for many ap- proximation algorithms in the literature, the error guar- Many of the computational problems that arise in antee is much higher (50%, 100%, 1000%, 10000%, or practice are optimization problems: the task is to find even worse). In this case we cannot argue that this ap- a solution where the cost, quality, size, profit, or some proximation algorithm is almost as good as an exact other measure is as large or small as possible. The algorithm. Nevertheless, such algorithms are still im- NP-hardness of an optimization problem implies that, portant from the theoretical point of view, as they al- unless P = NP, there is no polynomial-time algorithm low us to better understand and classify problems and that finds the exact value of the optimum. -
Bandwidth of Bipartite Permutation Graphs in Polynomial Time ∗
Bandwidth of bipartite permutation graphs in polynomial time ∗ Pinar Heggernes† Dieter Kratsch‡ Daniel Meister† Abstract We give the first polynomial-time algorithm that computes the bandwidth of bipartite permutation graphs. Bandwidth is an NP-complete graph layout problem that is notorious for its difficulty even on small graph classes. For example, it remains NP-complete on caterpillars of hair length at most 3, a very restricted subclass of trees. Much attention has been given to designing approximation algorithms for computing the bandwidth, as it is NP- hard to approximate the bandwidth of general graphs with a constant factor guarantee. The problem is considered important even for approximation on restricted classes, with several distinguished results in this direction. Prior to our work, polynomial-time algorithms for exact computation of bandwidth were known only for caterpillars of hair length at most 2, chain graphs, cographs, and most interestingly, interval graphs. 1 Introduction The bandwidth problem asks, given a graph G and an integer k, whether there exists a linear layout of the vertices of G such that no edge of G has its endpoints mapped to positions at distance more than k. The problem is motivated by sparse matrix computations, where given an n×n matrix A and an integer k, the goal is to decide whether there is a permutation matrix P such that P AP T is a matrix with all nonzero entries on the main diagonal or on the k diagonals on either side of the main diagonal. This problem is of great importance in many engineering applications since matrices arising in practice are often sparse. -
Crossing Minimization for 1-Page and 2-Page Drawings of Graphs with Bounded Treewidth
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth Michael J. Bannister and David Eppstein Department of Computer Science, University of California, Irvine Abstract. We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed- parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Cour- celle’s theorem on the fixed-parameter tractability of properties express- ible in monadic second order logic for graphs of bounded treewidth. 1 Introduction A k-page book embedding of a graph G is a drawing that places the vertices of G on a line (the spine of the book) and draws each edge, without crossings, inside one of k half-planes bounded by the line (the pages of the book) [16,19]. In one common drawing style, an arc diagram, the edges in each page are drawn as circular arcs perpendicular to the spine [24], but the exact shape of the edges is unimportant for the existence of book embeddings. These embeddings can be generalized to k-pagebookdrawings: as before, we place each vertex on the spine and each edge within a single page, but with crossings allowed. The crossing number of such a drawing is defined to be the sum of the numbers of crossings within each page, and the k-page crossing number crk(G) is the minimum number of crossings in any k-page book drawing [22].