Network Flows


Course Syllabus

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Course Objective

This is an advanced course on network models and algorithms in Operations research. The emphasis is on algorithm development and efficiency of algorithms. The follow-up course for this one is the course OPRE 7314: Combinatorial Optimization.

Office Hours

Jonsson 4.926
214-883-2032
TA: To be announced in Class

Text

  • Network Programming, by Katta G. Murty, Prentice Hall, 1992.
  • Network Flows, by R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Prentice Hall, 1993.

    Prerequisites

    OPRE 6310; or consent of the instructor.

    Grading Scheme

    Homework and two take-home exams

    Course Outline

    1. Graph Theory
      • Definitions and Concepts
    2. Matrices Associated with Graphs
      • Node-Arc Incidence Matrix
      • Arc Chain Incidence Matrix
      • The Loop or Mesh Matrix
      • The Node-Edge Incidence Matrix
      • The Cut-set Matrix
      • Orthogonolity
    3. Single Commodity Maximum Flow Problem
      • Formulation as an LP
      • Max-Flow-Min-Cut Theorem
      • Labeling Algorithm
      • Finite Termination of Maximum Flow Algorithm
        • Ford-Fulkerson Example
        • Queyranne Example
      • Strongly Polynomial Algorithms
        • Edmonds-Karp
        • Dinic
        • Karzanov
        • Maheshwari et al.
      • Special Cases
        • Undirected Networks
        • Parallel Arcs
        • Multiple Origins/Destinations
        • Node Capacities
      • Arc Chain Formulation and Chain Decomposition of Flows
      • Lower Bounds and Exogeneous Flows
      • Picard-Queyranne work and Structure of Cuts
      • MIT Work
    4. Single Commodity Multi-terminal Flows
    5. Spanning Tree Problem
    6. Shortest Path Problem
      • Structure
        • Specified Pair of Nodes
        • One Origin Several Destinations
        • All Pairs
        • Acyclic Graphs
        • Undirected Graphs
      • Data
        • Nonnegative Distances
        • Time Dependent Distances
        • Non-negative Cycles
      • Methods Used
        • Linear Programming Formulation
        • Dynamic Programming
        • Inductive Algorithm
        • Analog Devices
        • Matching Formulations
      • Objective Functions
        • Negative Cycle Detection
        • Ratio Functions
        • Ratio Functions on Cycles
        • kth Shortest Paths
        • Shortest Paths Through Specified Nodes
        • Odd and Even Shortest paths
      • Applications
        • PERT
        • Reliability
        • Ship Routing
        • Investments in Blocking Systems
    7. Minimum Cost Flows
      • Transportation and Primal Dual Algorithm
      • Out-of-kilter Algorithm
      • Klein's Algorithm
      • Edmonds-Karp Scaling
      • Zadeh's Examples
      • Strongly Polynomial Algorithms
        • E. Tardos
        • Fujishige
        • Simplex
        • Goldberg-Tarjan
        • PERT-CPM
    8. Unimodularity and Integrality
    9. Matching
      • Bipartite
      • General
      • Applications
        • Chinese Postman Problem
        • Scheduling
        • Odd and Even Shortest Paths
        • Shortest Paths in Undirected Networks with no Negative Cycles
        • Geometric Applications
        • Applications to Chemistry
      • Ellipsoid Algorithm and the Odd-Cut Problem
      • T-joins and T-cuts
    10. Blocking Systems and Polyhedra
      • Bottleneck Problems
      • Blocking Systems
      • Blocking and Antiblocking Polyhdera
      • Perfect Graph Theorem
    11. Multicommodity Flows
      • General Problem
      • Two Commodity
        • T.C. Hu's work
        • M. Sakarovitch
        • Rothschild and Whinston
      • H-Graphs
        • Single Star
        • Two Stars
        • Complete Graph and its Equivalents
        • Cut Based Problems
        • (2,3)-Metric Based Problems
    12. Matroids
      • Introduction
      • Equivalent Axioms
        • Independence
        • Circuits
        • Bases
        • Rank Function
        • Hyperplanes, flats etc.
        • Greedy Algorithm
      • Optimization Problems and Algorithms
        • Covering
        • Partition
        • Intersection
        • Parity
      • Polyhedral Charecterizations
      • Operations
        • Union
        • Deletion
        • Contraction
        • Truncation
        • Minors
        • Duals
        • Splitters
        • Decomposition and Composition
      • Applications
        • Testing Graphicness
        • Testing Regularity (Total Unimodularity)
        • Max-Flow-Min-Cut Matroids
      • Pseudomatroids
    13. Total Dual Integrality
      • General Theory
      • Integer Rounding
      • General Algorithm for Solving TDI Optimization Problems
      • Testing
      • Special Cases
      • Hilbert Basis
    14. Geometry of Numbers
      • Introduction
      • Lattice Basis Reduction Techniques
      • Lenstra's Algorithm
      • Scarf's Work
      • Tardos - Frank Simultaneous Approximation
      • Other Applications

    Assignments


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