Module Network Flows
- Network Flows Problem: introduction and definitions
- Maximum Flows and the path packing problem. Flows and cuts: Max-Flow/Min-Cut theorem. Augmenting path algorithms: Ford and Fulkerson algorithm, Edmonds and Karp algorithm. Generic Preflow-Push algorithm. Flows with lower bounds.
- Maximum Flows: additional topics and applications. Flows in Unit Capacity Networks. Flows in Bipartite Networks. Network Connectivity.
- Minimum Cuts. Global Minimum Cuts. Node Identification Algorithm. Random Contraction. Applications.
- Minimum-Cost Flow Problems. Definition and applications. Optimality Conditions. The Ford-Bellman algorithm for the shortest path problem. Primal algorithms: Augmenting Circuit Algorithm for the Min Cost Flow Problem.
- Network Simplex Algorithms. Applications of Min Cost Flows.
Module Network Optimization
- Formulations of Integer and Binary Programs: The Assignment Problem; The Stable Set Problem; Set Covering, Packing and Partitioning; Minimum Spanning Tree; Traveling Salesperson Problem (TSP); Formulations of logical conditions.
- Mixed Integer Formulations: Modeling Fixed Costs; Uncapacitated Facility Location; Uncapacitated Lot Sizing; Discrete Alternatives; Disjunctive Formulations.
- Optimality, Relaxation and Bounds. Geometry of R^n: Linear and affine spaces; Polyhedra: dimension, representations, valid inequalities, faces, vertices and facets; Alternative (extended) formulations; Good and Ideal formulations.
- LP based branch-and-bound algorithm: Preprocessing, Branching strategies, Node and variable selection strategies, Primal heuristics.
- Cutting Planes algorithms. Valid inequalities. Automatic Reformulation: Gomory’s Fractional Cutting Plane Algorithm. Strong valid inequalities: Cover inequalities, lifted cover inequalities; Clique inequalities; Subtour inequalities. Branch-and-cut algorithm.
- Software tools for Mixed Integer Programming.
- Lagrangian Duality: Lagrangian relaxation; Lagrangian heuristics.
- Network Problems: formulations and algorithms. Constrained Spanning Tree Problems; Constrained Shortest Path Problem; Multicommodity Flows; Symmetric and Asymmetric Traveling Salesman Problem; Vehicle Routing Problem; Steiner Tree Problem; Network Design.
- Heuristics for network problems: local search, tabu search, simulated annealing, MIP based heuristics.