Alzalg, Baha,
Combinatorial and algorithmic mathematics : from foundation to optimization / Baha Alzalg, the University of Jordan, Amman, Jordan. - 1 online resource (xxi, 506 pages) : illustrations (chiefly color) -
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"--
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.
9781394235940 139423595X 1394235968 1394235976 9781394235957 9781394235964 9781394235971
Algorithms.
Combinatorial analysis.
Combinatorial optimization.
Electronic books.
QA164 / .A4693 2024
511/.6
Combinatorial and algorithmic mathematics : from foundation to optimization / Baha Alzalg, the University of Jordan, Amman, Jordan. - 1 online resource (xxi, 506 pages) : illustrations (chiefly color) -
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"--
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.
9781394235940 139423595X 1394235968 1394235976 9781394235957 9781394235964 9781394235971
Algorithms.
Combinatorial analysis.
Combinatorial optimization.
Electronic books.
QA164 / .A4693 2024
511/.6