Combinatorial and Algorithmic Mathematics 1st Edition by Baha Alzalg ISBN 139423595X 9781394235957

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Product details: 

ISBN 10: 139423595X 

ISBN 13: 9781394235957

Author: Baha Alzalg

Supported by rigorous math and computer science foundations, Combinatorial and Algorithmic Mathematics: From Foundation to Optimization provides a from-scratch understanding to the field of optimization, discussing 70 algorithms with roughly 220 illustrative examples, 160 nontrivial end-of-chapter exercises with complete solutions to ensure readers can apply appropriate theories, principles, and concepts when required, and Matlab codes that solve some specific problems. This book helps readers to develop mathematical maturity, including skills such as handling increasingly abstract ideas, recognizing mathematical patterns, and generalizing from specific examples to broad concepts.

Starting from first principles of mathematical logic, set-theoretic structures, and analytic and algebraic structures, this book covers both combinatorics and algorithms in separate sections, then brings the material together in a final section on optimization. This book focuses on topics essential for anyone wanting to develop and apply their understanding of optimization to areas such as data structures, algorithms, artificial intelligence, machine learning, data science, computer systems, networks, and computer security.

Combinatorial and Algorithmic Mathematics includes discussion on:

• Propositional logic and predicate logic, set-theoretic structures such as sets, relations, and functions, and basic analytic and algebraic structures such as sequences, series, subspaces, convex structures, and polyhedra
• Recurrence-solving techniques, counting methods, permutations, combinations, arrangements of objects and sets, and graph basics and properties
• Asymptotic notations, techniques for analyzing algorithms, and computational complexity of various algorithms
• Linear optimization and its geometry and duality, simplex and non-simplex algorithms for linear optimization, second-order cone programming, and semidefinite programming

Combinatorial and Algorithmic Mathematics is an ideal textbook resource on the subject for students studying discrete structures, combinatorics, algorithms, and optimization. It also caters to scientists across diverse disciplines that incorporate algorithms and academics and researchers who wish to better understand some modern optimization methodologies.

Table of contents: 

Part I: Foundations
1 Mathematical Logic
1.1 Propositions
1.2 Logical Operators
1.3 Propositional Formulas
1.4 Logical Normal Forms
1.5 The Boolean Satisfiability Problem
1.6 Predicates and Quantifiers
1.7 Symbolizing Statements of the Form “All P Are Q”
Exercises
Notes and Sources
References
Notes
2 Set-Theoretic Structures
2.1 Induction
2.2 Sets
2.3 Relations
2.4 Partitions
2.5 Functions
Exercises
Notes and Sources
References
Note
3 Analytic and Algebraic Structures
3.1 Sequences
3.2 Summations and Series
3.3 Matrices, Subspaces, and Bases
3.4 Convexity, Polyhedra, and Cones
3.5 Farkas’ Lemma and Its Variants
Exercises
Notes and Sources
References
Notes
Part II: Combinatorics
4 Graphs
4.1 Basic Graph Definitions
4.2 Isomorphism and Properties of Graphs
4.3 Eulerian and Hamiltonian Graphs
4.4 Graph Coloring
4.5 Directed Graphs
Exercises
Notes and Sources
References
Notes
5 Recurrences
5.1 Guess-and-Confirm
5.2 Recursion-Iteration
5.3 Generating Functions
5.4 Recursion-Tree
Exercises
Notes and Sources
References
Notes
6 Counting
6.1 Binomial Coefficients and Identities
6.2 Fundamental Principles of Counting
6.3 The Pigeonhole Principle
6.4 Permutations
6.5 Combinations
Exercises
Notes and Sources
References
Notes
Part III: Algorithms
7 Analysis of Algorithms
7.1 Constructing and Comparing Algorithms
7.2 Running Time of Algorithms
7.3 Asymptotic Notation
7.4 Analyzing Decision-Making Statements
7.5 Analyzing Programs Without Function Calls
7.6 Analyzing Programs with Function Calls
7.7 The Complexity Class NP-Complete
Exercises
Notes and Sources
References
Notes
8 Array and Numeric Algorithms
8.1 Array Multiplication Algorithms
8.2 Array Searching Algorithms
8.3 Array Sorting Algorithms
8.4 Euclid’s Algorithm
8.5 Newton’s Method Algorithm
Exercises
Notes and Sources
References
Note
9 Elementary Combinatorial Algorithms
9.1 Graph Representations
9.2 Breadth-First Search Algorithm
9.3 Applications of Breadth-First Search
9.4 Depth-First Search Algorithm
9.5 Applications of Depth-First Search
9.6 Topological Sort
Exercises
Notes and Sources
References
Note
Part IV: Optimization
10 Linear Programming
10.1 Linear Programming Formulation and Examples
10.2 The Graphical Method
10.3 Standard Form Linear Programs
10.4 Geometry of Linear Programming
10.5 The Simplex Method
10.6 Duality in Linear Programming
10.7 A Homogeneous Interior-Point Method
Exercises
Notes and Sources
References
Notes
11 Second-Order Cone Programming
11.1 The Second-Order Cone and Its Algebraic Structure
11.2 Second-Order Cone Programming Formulation
11.3 Applications in Engineering and Finance
11.4 Duality in Second-Order Cone Programming
11.5 A Primal-Dual Path-Following Algorithm
11.6 A Homogeneous Self-Dual Algorithm
Exercises
Notes and Sources
References
Notes
12 Semidefinite Programming and Combinatorial Optimization
12.1 The Cone of Positive Semidefinite Matrices
12.2 Semidefinite Programming Formulation
12.3 Applications in Combinatorial Optimization
12.4 Duality in Semidefinite Programming
12.5 A Primal–Dual Path-Following Algorithm
Exercises
Notes and Sources
References
Appendix A: Solutions to Chapter ExercisesSolutions to Chapter Exercises
References
Note
Bibliography
Index
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