Combinatorial and algorithmic mathematics : from foundation to optimization / Baha Alzalg, the University of Jordan, Amman, Jordan.

Includes bibliographical references and index.

Table of Contents
About the Author xiii

Preface xv

Acknowledgments xvii

About the Companion Website xxi

Part I Foundations 1

1 Mathematical Logic 3

1.1 Propositions 3

1.2 Logical Operators 6

1.3 Propositional Formulas 15

1.4 Logical Normal Forms 24

1.5 The Boolean Satisfiability Problem 29

1.6 Predicates and Quantifiers 30

1.7 Symbolizing Statements of the Form "All P Are Q" 37

2 Set-Theoretic Structures 51

2.1 Induction 51

2.2 Sets 54

2.3 Relations 59

2.4 Partitions 64

2.5 Functions 65

3 Analytic and Algebraic Structures 77

3.1 Sequences 77

3.2 Summations and Series 81

3.3 Matrices, Subspaces, and Bases 87

3.4 Convexity, Polyhedra, and Cones 91

3.5 Farkas' Lemma and Its Variants 95

Part II Combinatorics 103

4 Graphs105

4.1 Basic Graph Definitions 106

4.2 Isomorphism and Properties of Graphs 113

4.3 Eulerian and Hamiltonian Graphs 118

4.4 Graph Coloring 122

4.5 Directed Graphs 125

5 Recurrences 133

5.1 Guess-and-Confirm 133

5.2 Recursion-Iteration 136

5.3 Generating Functions 138

5.4 Recursion-Tree 140

6 Counting149

6.1 Binomial Coefficients and Identities 149

6.2 Fundamental Principles of Counting 154

6.3 The Pigeonhole Principle 161

6.4 Permutations 163

6.5 Combinations 166

Part III Algorithms 179

7 Analysis of Algorithms 181

7.1 Constructing and Comparing Algorithms 182

7.2 Running Time of Algorithms 189

7.3 Asymptotic Notation 199

7.4 Analyzing Decision-Making Statements 211

7.5 Analyzing ProgramsWithout Function Calls 213

7.6 Analyzing Programs with Function Calls 219

7.7 The Complexity Class NP-Complete 224

8 Array and Numeric Algorithms 241

8.1 Array Multiplication Algorithms 241

8.2 Array Searching Algorithms 244

8.3 Array Sorting Algorithms 248

8.4 Euclid's Algorithm 253

8.5 Newton's Method Algorithm 255

9 Elementary Combinatorial Algorithms 267

9.1 Graph Representations 267

9.2 Breadth-First Search Algorithm 270

9.3 Applications of Breadth-First Search 273

9.4 Depth-First Search Algorithm 277

9.5 Applications of Depth-First Search 279

9.6 Topological Sort 283

Part IV Optimization 293

10 Linear Programming 295

10.1 Linear Programming Formulation and Examples 296

10.2 The Graphical Method 302

10.3 Standard Form Linear Programs 309

10.4 Geometry of Linear Programming 311

10.5 The Simplex Method 320

10.6 Duality in Linear Programming 339

10.7 A Homogeneous Interior-Point Method 347

11 Second-Order Cone Programming 363

11.1 The Second-Order Cone and Its Algebraic Structure 363

11.2 Second-Order Cone Programming Formulation 368

11.3 Applications in Engineering and Finance 370

11.4 Duality in Second-Order Cone Programming 375

11.5 A Primal-Dual Path-Following Algorithm 379

11.6 A Homogeneous Self-Dual Algorithm 386

12 Semidefinite Programming and Combinatorial Optimization 395

12.1 The Cone of Positive Semidefinite Matrices 395

12.2 Semidefinite Programming Formulation 399

12.3 Applications in Combinatorial Optimization 401

12.4 Duality in Semidefinite Programming 405

12.5 A Primal–Dual Path-Following Algorithm 408

Exercises 417

Notes and Sources 418

References 418

Appendix A Solutions to Chapter Exercises 421

References 487

Bibliography 489

Index 501

"Throughout operations research, computer science and pure and applied mathematics, combinatorics problems arise frequently, where the solution is to find the "optimal" object from a finite set of mathematical objects. Typically, it is impractical to search exhaustively for all possible solutions. The development of efficient algorithms for exploring the solution space is known as combinatorial optimisation. Many problems, such as network optimisation, supply chain management, data compression, resource allocation and game theory - indeed most of machine learning, AI, and current high profile computer science topics rely on optimisation. Together, combinatorial and algorithmic mathematics provide powerful tools for solving these real-world problems, and for in-demand subjects such as data science, machine learning, and artificial intelligence, a unified knowledge of discrete structures, algorithms and combinatorial optimization is considered essential"-- Provided by publisher.

About the Author
Baha Alzalg is a Professor in the Department of Mathematics at the University of Jordan in Amman, Jordan. He has also held the post of visiting associate professor in the Department of Computer Science and Engineering at the Ohio State University in Columbus, Ohio. His research interests include topics in optimization theory, applications, and algorithms, with an emphasis on interior-point methods for cone programming.