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Analysis of Algorithms for Star Bicoloring and Related Problems A Analysis of Algorithms for Star Bicoloring and Related Problems A dissertation presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Jeffrey S. Jones May 2015 © 2015 Jeffrey S. Jones. All Rights Reserved. 2 This dissertation titled Analysis of Algorithms for Star Bicoloring and Related Problems by JEFFREY S. JONES has been approved for the School of Electrical Engineering and Computer Science and the Russ College of Engineering and Technology by David Juedes Professor of Electrical Engineering and Computer Science Dennis Irwin Dean, Russ College of Engineering and Technology 3 Abstract JONES, JEFFREY S., Ph.D., May 2015, Computer Science Analysis of Algorithms for Star Bicoloring and Related Problems (171 pp.) Director of Dissertation: David Juedes This dissertation considers certain graph-theoretic combinatorial problems which have direct application to the efficient computation of derivative matrices “Jacobians”) which arise in many scientific computing applications. Specifically, we analyze algorithms for Star Bicoloring and establish several analytical results. We establish complexity-theoretic lower bounds on the approximability of algorithms for Star Bicoloring, showing that no such polynomial-time algorithm can achieve an 1 ǫ approximation ratio of O(N 3 − ) for any ǫ > 0 unless P = NP. We establish the first algorithm (ASBC) for Star Bicoloring with a known approximation upper-bound, 2 showing that ASBC is an O(N 3 ) polynomial-time approximation algorithm. Based on extension of these results we design a generic framework for greedy Star Bicoloring, and implement several specific methods for comparison. General analysis techniques are developed and applied to both algorithms from the literature (CDC, Hossain and Steihaug, 1998 [1]) as well as those developed as part of the framework. We provide numerous approximability results including the first approximation analysis for the CDC algorithm, 3 showing that CDC is an O(N 4 ) approximation algorithm. Finally, we observe that all algorithms within this generic framework produce a restricted class of star bicolorings that we refer to as Distance-2 Independent Set (D2IS C) colorings. We establish the relationship between Star Bicoloring and D2IS C. In particular we show that these two notions are not equivalent, that D2IS C is NP-complete and that it cannot be approximated 1 ǫ to within O(N 3 − ) for any ǫ > 0 unless P = NP. 4 This work is dedicated to the memory of my parents, Ora and Violet Jones, who were the first in their family to ensure that their children enjoyed the benefits of higher education. This work is dedicated to my wife Janis, son Gordon and daughter Margaret, for their gracious support and encouragement. I have “kept my eye on the hat.” This work is dedicated to my brother Bill, who has been an inspiration and guide to me throughout my life. 5 Acknowledgements I would like to extend sincere thanks to Dr. David Juedes for his instruction, support, collaboration, guidance and friendship throughout my entire Ph.D. experience. Dr. Juedes has a gift for algorithmic analysis, a contagious joy for collaborative pursuit of the field, and a cheerful willingness to pass along his knowledge. I would like to thank the members of my Ph.D. dissertation committee, who have given their time both in and outside of the classroom in support of my candidacy: Dr. Razvan Bunescu; Dr. Frank Drews; Dr. Cynthia Marling; Dr. Sergio Lopez; and Dr. Howard Dewald. 6 Table of Contents Page Abstract ......................................... 3 Dedication ........................................ 4 Acknowledgements ................................... 5 List of Tables ...................................... 9 List of Figures ...................................... 10 List of Symbols ..................................... 12 List of Acronyms .................................... 14 1 Introduction ..................................... 16 1.1 Overview ................................... 18 1.2 Background, Motivation and Research Direction ............... 20 1.3 Examples ................................... 23 1.3.1 Combustion of Propane ........................ 23 1.3.2 Curtis54 ................................ 25 1.3.3 A Representative Sparsity Pattern .................. 28 2 Background and Literature Review ......................... 30 2.1 Preliminary Mathematical Notation ..................... 30 2.2 Applications of the Jacobian ......................... 31 2.3 Sparse Derivative Matrices and Sparsity Patterns .............. 34 2.4 The Connection between Derivative Matrices and Graphs .......... 35 2.5 Generic Coloring Approaches ........................ 36 2.6 D-1 Coloring Methods ............................ 38 2.7 Orthogonal Independence ........................... 39 2.8 D-2 Coloring Methods ............................ 40 2.9 Uni-directional versus Bicoloring Approaches ................ 42 2.10 Bidirectional Direct Computation, SeqDC .................. 44 2.11 Chromatic and Coloring Numbers ...................... 45 2.12 Direct versus Substitution Methods ...................... 47 2.13 Ordering Methods ............................... 48 2.13.1 Largest First ............................. 48 2.13.2 Smallest Last ............................. 48 2.13.3 Incidence Degree ........................... 49 7 2.13.4 Saturation Degree ........................... 49 2.14 Existing Coloring Algorithms ......................... 50 2.14.1 uni-directional coloring ........................ 50 2.14.2 Minimum Nonzero Count Ordering with Incidence Degree Order- ing (MNCO/IDO) ........................... 50 2.14.3 Complete Direct Cover (CDC) .................... 52 3 Initial Results .................................... 54 3.1 An Approximation Lower Bound ....................... 54 3.2 Approximate Star BiColoring (ASBC), an Approximation Algorithm . 56 3.3 ASBC Approximation Analysis ....................... 60 3.3.1 Edge Elimination ........................... 60 3.3.2 Limit on Colors Used ......................... 62 3.3.3 Approximation Ratio ......................... 62 3.4 ASBC Comparative Empirical Results .................... 63 4 A Family of Greedy Star Bicoloring Algorithms .................. 65 4.1 A General Star Bicoloring Framework .................... 65 4.2 A Class for GIS Algorithms ......................... 67 4.3 Selected Greedy Strategies, Part One ..................... 68 4.3.1 Locking, a Boundary Condition for Certain GIS Methods . 70 4.4 Selected Greedy Strategies, Part Two ..................... 73 4.4.1 A Limitation to the Neighborhood Ratio Method .......... 79 4.5 Non-greedy Strategies ............................ 81 4.6 Empirical Results ............................... 83 5 Expanded Analyses ................................. 89 5.1 Correctness of General Framework ...................... 90 5.2 An Improved Lower Bound for GIS Family Methods Using Seed Vertex of Maximum Degree (MDGIS) ......................... 92 5.3 Observations on the Size of GIS Neighborhoods ............... 94 5.4 CDC Approximation Analysis ........................ 95 5.5 CDC Approximation Based on E ...................... 97 5.6 Initial Maximum Neighborhood| Approximation| Analysis .......... 99 5.7 Improved Maximum Neighborhood Approximation Analysis ........100 5.8 Approximation Analyses for Almost Square Matrices ............101 5.8.1 Hypothetical Strongly Square Matrices ...............101 5.8.2 Nearly Square Matrices ........................103 5.8.2.1 An Example Nearly Square Matrix Construction . 104 5.9 Neighborhood Ratio Approximation Analysis ................107 5.10 Analysis for Ratio M’ Method ........................109 5.11 Analysis for Inverse Ratio Method ......................110 5.12 Analysis for Dense Ratio Method .......................111 8 5.13 Look Ahead Approximation Analysis ....................112 5.14 Weighted Unlocking Approximation Analysis ................118 5.15 Analysis Summary ..............................120 5.15.1 Observations .............................122 6 Distance-2 Independent Set Coloring ........................124 6.1 The Inequivalence of D2IS Bicoloring and Star Bicoloring .........125 6.2 NP-completeness for D2IS Coloring .....................127 6.2.1 Graph Coloring to Distance-2 Independent Set Coloring Reduction Construction .............................128 6.2.2 Graph Coloring Distance-2 Independent Set Coloring . 128 6.2.3 Distance-2 Independent→ Set Coloring Graph Coloring . 129 → 7 Concluding Thoughts and Future Directions ....................131 7.1 Algorithmic Directions ............................132 7.2 Further Analyses ...............................133 7.3 Other Considerations .............................134 References ........................................135 Appendix: The Matrix M class .............................139 9 List of Tables Table Page 3.1 Comparative Coloring Performance [2] ..................... 64 4.1 “Strategy T” Variations .............................. 74 4.2 Full List of GIS Evaluation Test Matrices .................... 84 4.3 Benchmark Coloring Results - Greedy Methods ................. 87 4.4 Benchmark Coloring Results - “Non-greedy” Methods ............. 88 5.1 GIS Methods, Analysis Summary ........................123 10 List of Figures Figure Page 1.1 Coefficient Matrix for the Combustion of Propane System of Equations . 24 1.2 Sparsity Pattern for the Combustion of Propane System of Equations . 24 1.3 Color Groups for the
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