Linear orderings of sparse graphs:
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| Format: | Thesis Book |
| Language: | English |
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Passau
2017
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| Online Access: | Volltext Volltext Inhaltsverzeichnis |
| Physical Description: | vii, 282 Seiten Illustrationen, Diagramme |
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| 245 | 1 | 0 | |a Linear orderings of sparse graphs |c Kathrin Hanauer |
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Record in the Search Index
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| adam_text | 1 Introduction
Contents
1
2 The Linear Ordering Problem: An Outline 5
2.1 The Linear Ordering Problem and its Kin........................... 6
2.1.1 Problem Statements ...................................... 6
2.1.2 Linear Programming ....... . . .............. ... ........ 9
2.1.3 Dual Problems ............................................... 11
2.1.4 The Acyclic Subgraph Polytope ............................... 13
2.1.5 Reductions to and from the LINEAR ORDERING Problem ..... 14
2.2 Complexity........................................................ 18
2.2.1 .A/’T’-Completeness Results ................................ 18
2.2.2 Approximability and Approximations .................... . 21
2.2.3 Parameterized Complexity..................................... 23
2.2.4 Polynomially Solvable Instances..................... 25
2.3 The Cardinality of Optimal Feedback Arc Sets....................... 26
2.3.1 Existential Bounds.....................................!... 26
2.3.2 Algorithms with Absolute Performance Guarantees . . ........ 27
2.4 Linear Orderings in Practice ..................................... 28
2.4.1 Heuristics . .... 1 ... ............................... 28
2.4.2 Applications ............................................. 30
3 Preliminaries: Definitions and Preparations 35
3.1 Basics........................................................... 35
3.1.1 Sets and Multisets ................................... . 35
3.1.2 Minimal and Maximal versus Minimum and Maximum . . ... 37
3.1.3 Computational Complexity................... 37
3.1.4 Data Structures............................... 37
3.2 General Graph Theory................................ . . ..... 38
3.2.1 Graphs, Vertices, and Arcs ......... ... . ... . . . . ... . 38
IV
Contents
3.2.2 Vertex Degrees................................................ 39
3.2.3 Paths, Cycles, and Walks.................................... 40
3.2.4 Connectivity and Acyclicity................................... 41
3.3 Feedback Arc Sets and Linear Orderings.............................. 43
3.3.1 Feedback Arc Sets ........... . ............................ 43
3.3.2 Linear Orderings.............................................. 44
3.3.3 Forward Paths and Layouts..................................... 47
3.4 Preprocessing and Default Assumptions............................... 48
3.4.1 Loops and Anti-Parallel Arcs.................................. 48
3.4.2 Strong Connectivity.......................................... 49
3.4.3 Biconnectivity.............................................. 50
4 Properties of Optimal Linear Orderings: A Microscopic View 55
4.1 General Framework................................. . . ........... 55
4.2 Algorithmic Setup ................................................ 58
4.2.1 Graphs........................................................ 58
4.2.2 Linear Orderings . ......................................... 59
4.2.3 Vertex Layouts................................................ 59
4.2.4 Initializing the Data Structures............................ 60
4.2.5 General Remarks............................................... 63
4.3 Nesting Property................................................. 63
4.3.1 A 1-opt Algorithm............................................. 63
4.3.2 Nesting Arcs ................................................. 68
4.3.3 A Graph s Excess............................................ 71
,4.4 Path Property....................................................... 73
4.4.1 Forward Paths for Backward Arcs............................... 73
4.4.2 Establishing Forward Paths.................................. 74
4.4.3 Minimal Feedback Arc Sets .................................... 76
- 4.5 Blocking Vertices Property........................................ 77
4.5.1 Left- and Right-Blocking Vertices........................... 78
4.5.2 Vertical Splits ............................................ 82
4.5.3 Establishing Non-Blocking Forward Paths....................... 84
4.6 Multipath Property.................................................. 87
4.6.1 Arc-Disjoint Forward Paths................................... 87
4.6.2 Analyzing the Flow Network Approach........................... 91
Contents v
4.6.3 Arc-Disjoint Cycles ... . . . . ................. . . . . . . . 94
4.6.4 An AfV-hard. Extension .................. 95
4.7 Multipath Blocking Vertices Property . .,....■........ .......... 97
4.7.1 Non-Blocking Multipaths ........................ 98
4.7.2 Flow Networks for Split Graphs....................... 103
4.7.3 Again an A/’P-hard Extension ......................... 107
4.8 Eliminable Layouts Property ..................................... 112
4.8.1 Eliminable Layouts............................ . . . . 112
4.8.2 The Elimination Operation . ..............................117
4.8.3 Eliminating Eliminable Layouts . :............ .......... 122
4.9 A PsiOpt-Algorithm . ........................................... 126
4.9.1 A Cascading Meta-Algorithm ............................ 127
4.9.2 Establishing the Necessary Properties Simultaneously . . . i . . 129
4.10 Manipulations and Meta-Properties............ 132
4.10.1 Basic Operations on Linear Orderings and Graphs........... 133
4.10.2 Fusion Property..................................... 143
4.10.3 Reduction Property ..............: . . 146
4.10.4 Arc Stability Property .................... 149
5 Maximum Cardinality of Optimal Feedback Arc Sets of Sparse Graphs 151
5.1 Auxiliary Graphs......................................... 152
5.1.1 The Forward Path Graph.................. . .......:..... 152
5.1.2 The Pooled Forward Path Graph ............................ 154
5.1.3 The Polarized Forward Path Graph ....................; 156
5.1.4 The Truncated Forward Path Graph ............. i. ;. . 157
5.2 Subcubic Graphs................................................ . . 159
5.2.1 A Tight Bound ............................................ 159
5.2.2 On the Approximation Ratio.............................. 1 . 163
5.3 From Subcubic to General Graphs ............... . . .:........ 167
5.3.1 Pebble Transportation in Supercubic Graphs.......... . . . 168
5.3.2 A General Assignment Scheme................................ 169
5.4 Subquartic and Subquintic Graphs................... 173
5.4.1 Two Special Cases of One-Arc Stability..................... 173
5.4.2 A Tight Bound for Subquartic Graphs........................ 177
5.4.3 Subquintic Graphs......................................... 193
Contents
vi-
6 Exact and Fast Algorithms for- Linear Ordering 197
6.1 Partial Layouts and Incomplete Linear Orderings . -.............. 198
6.2 Exact Algorithms for Optimization and Decision ...................201
• 6.3 Branch and Bound with Integrated Partial Layouts . . . .......... 208
6.4 Fine-Tuning.................................................... 218
6.5 Runtime Comparison for Sparse Graphs 219
7 , Experimental Evaluation 221
7.1 The Algorithm Test Suite........................................ 222
7.1.1 Algorithms ............. . . . ......................... 222
7.1.2 Input Instances.......................................... 224
7.1.3 Technical Setup........................................ 226
7.1.4 Evaluation . ............................................ 227
7.2 Sparse Regular Graphs...........,............................... 228
7.2.1 Selection and Configuration of Algorithms................ 228
7.2.2 Performances and Running Times........................... 229
7.2.3 Summary................................................ 237
7.3 Large Graphs . . ............................................... 239
7.3.1 Fences, Ladders, and their Composites.................... 239
7.3.2 Performances and Running Times .......................... 240
i 7.3.3 Summary................................................... 243
7.4 The LOLIB Graph Library....................................... 244
7.4.1 Sets of LOLIB Instances . . . ............................244
. 7.4.2 Performances and Running Times......................... 246
: . 7.4.3 Comparison to Other Approaches........................... 249
7.5 Threats to Validity............................................. . 249
v 7.5.1 . Construct Validity...................................... 249
7.5.2 Internal Validity.......... . . ........................ 250
7.5.3 External Validity......................................... 250
7.6 Summary...................................................... 250
8 Conclusion and Future Work 253
Bibliography 255
Notation Index
267
Contents
vu
Algorithm Index
273
Subject Index
277
|
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| spelling | Hanauer, Kathrin Verfasser aut Linear orderings of sparse graphs Kathrin Hanauer Passau 2017 vii, 282 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Dissertation Universität Passau 2018 Schwach besetzte Matrix (DE-588)4056053-3 gnd rswk-swf Graphentheorie (DE-588)4113782-6 gnd rswk-swf Lineares Ordnungsproblem (DE-588)4120676-9 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Lineares Ordnungsproblem (DE-588)4120676-9 s Graphentheorie (DE-588)4113782-6 s Schwach besetzte Matrix (DE-588)4056053-3 s DE-604 Erscheint auch als Online-Ausgabe urn:nbn:de:bvb:739-opus4-5524 https://nbn-resolving.org/urn:nbn:de:bvb:739-opus4-5524 Resolving-System kostenfrei Volltext https://opus4.kobv.de/opus4-uni-passau/frontdoor/index/index/docId/552 kostenfrei Volltext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030476715&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hanauer, Kathrin Linear orderings of sparse graphs Schwach besetzte Matrix (DE-588)4056053-3 gnd Graphentheorie (DE-588)4113782-6 gnd Lineares Ordnungsproblem (DE-588)4120676-9 gnd |
| subject_GND | (DE-588)4056053-3 (DE-588)4113782-6 (DE-588)4120676-9 (DE-588)4113937-9 |
| title | Linear orderings of sparse graphs |
| title_auth | Linear orderings of sparse graphs |
| title_exact_search | Linear orderings of sparse graphs |
| title_full | Linear orderings of sparse graphs Kathrin Hanauer |
| title_fullStr | Linear orderings of sparse graphs Kathrin Hanauer |
| title_full_unstemmed | Linear orderings of sparse graphs Kathrin Hanauer |
| title_short | Linear orderings of sparse graphs |
| title_sort | linear orderings of sparse graphs |
| topic | Schwach besetzte Matrix (DE-588)4056053-3 gnd Graphentheorie (DE-588)4113782-6 gnd Lineares Ordnungsproblem (DE-588)4120676-9 gnd |
| topic_facet | Schwach besetzte Matrix Graphentheorie Lineares Ordnungsproblem Hochschulschrift |
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