Here are 100 chapter titles for a book on Integer Programming, progressing from beginner to advanced:
I. Foundations and Linear Programming (20 Chapters)
- Introduction to Optimization
- Linear Programming: Formulation and Graphical Solution
- The Simplex Method: Tableau Form
- The Simplex Method: Matrix Form
- Duality in Linear Programming
- Sensitivity Analysis and Parametric Programming
- The Transportation Problem
- The Assignment Problem
- Network Flow Problems: Max-Flow
- Network Flow Problems: Min-Cost Flow
- Introduction to Integer Programming: Formulations
- Types of Integer Programming Problems: Pure, Mixed, Binary
- Applications of Integer Programming: Real-World Examples
- Relaxations of Integer Programs: Linear Relaxation
- Rounding Techniques and Their Limitations
- Unimodularity and Total Unimodularity
- Introduction to Complexity Theory: P vs. NP
- Integer Programming and Complexity
- Software for Integer Programming: CPLEX, Gurobi, etc.
- Basic Modeling Techniques for Integer Programs
II. Solution Methods: Cutting Planes and Branch-and-Bound (30 Chapters)
- Cutting Plane Methods: Introduction
- Gomory Cuts: Fractional Cuts
- Gomory Cuts: All-Integer Cuts
- Cutting Plane Methods: Convergence and Finiteness
- Lift-and-Project Methods
- Branch-and-Bound: The Basic Algorithm
- Branching Strategies: Depth-First, Breadth-First, Best-Bound
- Node Selection and Exploration
- Fathoming: Primal Bounds, Dual Bounds, Infeasibility
- Cutting Planes within Branch-and-Bound: Branch-and-Cut
- Preprocessing and Problem Reduction
- Strong Formulations and Valid Inequalities
- Valid Inequalities for Specific Problems: Knapsack, Set Covering
- Generating Valid Inequalities: Lifting
- Polyhedral Theory and Integer Programming
- Faces, Facets, and Extreme Points
- The Cutting Stock Problem: A Case Study
- The Traveling Salesperson Problem (TSP): Formulations
- The Traveling Salesperson Problem: Cutting Planes
- The Traveling Salesperson Problem: Branch-and-Bound
- Set Covering and Set Partitioning Problems
- Facility Location Problems: Formulations
- Facility Location Problems: Solution Methods
- Scheduling Problems: Formulations and Solution Approaches
- Production Planning Problems
- Resource Allocation Problems
- Capital Budgeting Problems
- Combinatorial Optimization and Integer Programming
- Approximation Algorithms for Integer Programs (Introduction)
- Heuristics for Integer Programming
III. Advanced Topics and Decomposition (30 Chapters)
- Dantzig-Wolfe Decomposition
- Benders Decomposition
- Column Generation
- Lagrangian Relaxation and Duality
- Subgradient Optimization
- Cutting Plane Generation Techniques: Advanced
- Branching Strategies: Advanced Techniques
- Branch-and-Cut Algorithms: Advanced Implementations
- Computational Integer Programming: Software and Performance
- Parallel Algorithms for Integer Programming
- Distributed Algorithms for Integer Programming
- Robust Optimization and Integer Programming
- Stochastic Integer Programming
- Two-Stage Stochastic Programming
- Chance-Constrained Programming
- Integer Programming under Uncertainty
- Logic-Based Benders Decomposition
- Constraint Programming and Integer Programming
- Hybrid Methods: Combining Techniques
- Decomposition Methods for Large-Scale Integer Programs
- Reformulation-Linearization Techniques (RLT)
- Disjunctive Programming
- Extended Formulations
- Lifted Linear Programs
- Semidefinite Programming Relaxations
- Applications in Logistics and Supply Chain Management
- Applications in Finance
- Applications in Telecommunications
- Applications in Energy and Sustainability
- Applications in Healthcare
IV. Further Explorations and Specialized Topics (20 Chapters)
- Polyhedral Combinatorics
- Integer Programming and Graph Theory
- Matroid Theory and Integer Programming
- Submodular Functions and Integer Programming
- Online Integer Programming
- Dynamic Programming and Integer Programming
- Parameterized Complexity and Integer Programming
- Approximation Algorithms for Integer Programs: Advanced Topics
- Heuristics and Metaheuristics for Integer Programming: Advanced Topics
- Machine Learning and Integer Programming
- Data-Driven Integer Programming
- Integer Programming and Game Theory
- Integer Programming and Mechanism Design
- Integer Programming and Combinatorial Auctions
- Integer Programming in Bioinformatics
- Integer Programming in Computer Vision
- The History of Integer Programming
- Open Problems in Integer Programming
- Future Directions in Integer Programming
- Appendix: Foundational Material and References