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Computational Aspects of Treewidth ——————————————————————————————————— Lower Bounds and Network Reliability

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Computational Aspects of Treewidth ——————————————————————————————————— Lower Bounds and Network Reliability Computational Aspects of Treewidth ——————————————————————————————————— Lower Bounds and Network Reliability Computational Aspects of Treewidth ——————————————————————————————————— Lower Bounds and Network Reliability Computationele Aspecten van Boombreedte Ondergrenzen en Betrouwbaarheid van Netwerken (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. Dr. W. H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op maandag 13 juni 2005 des middags te 12.45 uur door Thomas Wolle geboren op 29 oktober 1975 te Gera promotor: Prof. Dr. Jan van Leeuwen Faculty of Science Utrecht University copromotor: Dr. Hans L. Bodlaender Faculty of Science Utrecht University The work in this thesis (IPA Dissertation series number 2005-09) has been carried out under the auspices of the research school IPA (Institute for Programming research and Algorithmics). This research was partially supported by EC contract IST-1999-14186: Project ALCOM-FT (Algorithms and Complexity - Future Technologies) and partially by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation). Printed by Ridderprint offsetdrukkerij B.V. ISBN 90-393-3972-4 Copyright c Thomas Wolle, 2005 Contents Preface . 1 1 Introduction . 3 1.1 Treewidth . 4 1.2 Motivation . 7 1.3 Thesis Outline . 9 2 Preliminaries . 11 2.1 General . 11 2.2 Edge Contraction . 12 2.3 Subgraphs and Minors . 17 2.4 Treewidth . 18 2.5 Subdividing an Edge . 19 3 Treewidth Lower Bounds: Methods, Relations and Complexity . 21 3.1 Contraction Degeneracy and Related Lower Bounds . 21 3.1.1 Definition of the Parameters . 22 3.1.2 Relationships Between the Parameters . 23 3.1.3 NP -completeness Results . 26 3.1.4 Fixed Parameter Case of CONTRACTION DEGENERACY . 33 3.1.5 On the Contraction Degeneracy . 35 3.1.6 Graphs of Genus γ . 39 3.2 Maximum Cardinality Search Lower Bound . 44 3.2.1 MCS Treewidth Lower Bound with Contraction . 45 3.2.2 NP -completeness . 45 3.2.3 Fixed Parameter Case . 48 3.3 Improved Graphs Might Improve Lower Bounds . 48 3.4 Concluding Remarks . 49 4 Treewidth Lower Bounds: Algorithms, Heuristics and Experiments . 51 4.1 A Bucket Adjacency-List Data Structure . 51 4.2 Exact Algorithms for Some Treewidth Lower Bounds . 55 4.2.1 Algorithms for δ, δ2 and δD . 55 4.2.2 Algorithms for γR . 55 4.2.3 An Algorithm for δ2D . 56 VI Contents 4.3 Heuristics . 57 4.3.1 δC-Heuristics . 57 4.3.2 δ2C-Heuristics . 61 4.3.3 γRD-Heuristics . 61 4.3.4 γRC-Heuristics . 62 4.3.5 MCSLB and MCSLBC-Heuristics . 62 4.3.6 LBN and LBP Heuristics . 62 4.4 Experiments . 65 4.4.1 Experimental Setup and Input Graphs . 66 4.4.2 Results for δ, δ2, γR, δD, δ2D and γRD . 66 4.4.3 Results for δC . 67 4.4.4 Results for δ2C and γRC . 68 4.4.5 Results for MCSLB and MCSLBC . 68 4.4.6 Results for the LBN and LBP Heuristics . 69 4.5 Concluding Remarks . 70 5 Contraction Degeneracy on Cographs . 73 5.1 Cographs . 73 5.2 Computing Contraction Degeneracy . 74 5.2.1 A Recurrence Relations for 0-nodes . 76 5.2.2 A Recurrence Relations for 1-nodes . 77 5.2.3 The Dynamic-Programming Method . 82 5.3 Concluding Remarks . 83 6 Network Reliability: Model, Problems and Complexity . 85 6.1 The Classical Network Reliability Problems . 85 6.2 Our Network Reliability Problems . 86 6.2.1 Our Graph-theoretical Model . 86 6.2.2 Edge Failures vs. Vertex Failures . 86 6.2.3 A List of Network Reliability Problems . 87 6.3 Complexity of Network Reliability Problems . 87 6.3.1 Complexity Theory Background . 88 6.3.2 #P 0-membership . 89 6.3.3 #P 0-hardness . 91 6.3.4 Computing the Numerators of Probabilities . 94 6.4 Concluding Remarks . 94 7 Network Reliability: Graphs of Bounded Treewidth . 97 7.1 A General Technique . ..
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