Discrete Optimization

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Discrete Optimization
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The relationship between counting
techniques/graph theory and discrete optimization

• Adding a goal (objective function) to a counting
  situation or a graph model makes it a discrete
  optimization problem.

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Example making the scheduling model an
optimization problem

• There are 4 jobs that should be processed
• on the same machine.
• (Cant be processed simultaneously).
• Here is an example of a possible schedule
• Original (counting) question What is the number
  of all possible schedules?
• New (optimization) question Find a schedule that
  minimizes the average completion time of the four
  jobs.

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Example making a graph model an optimization
problem

• There are n cities. The salesman
• ? starts his tour from City 1,
• ? visits each of the cities exactly once,
• ? and returns to City 1.
• Original (counting) question How many different
  tours are possible?
• New (optimization) question Find a minimum-cost
  tour.

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Graphs and discrete optimization

• Adding a goal (objective function) about the
  amount of some entity in a network makes the
  network model a discrete (network) optimization
  problem.
• Goal Build a network satisfying certain
  requirements with minimal cost
• Move some entity (electricity, a consumer
  product, people, information) from one point to
  another in underlying network as efficiently as
  possible
• Provide good service to the network users
• Use the network facilities efficiently

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Ingredients of some common physical networks
Application Physical analog of nodes Physical analog of arcs Flow
Transportation systems Cities, intersections, facilities Highways, airline routes Vehicles, passengers
Communication systems Phones, computers Cables, fiber optic links Voice messages, data
Hydraulic systems Pumping stations, reservoirs Pipelines Water, gas, oil
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How to solve discrete optimization problems?

• Design algorithms!
• The word algorithm refers to a step-by-step
  method for performing some action.
• Some examples of algorithms in everyday life
• Food preparation recipes
• Driving directions
• Directions for assembling equipment
• Instructions for filling out income tax forms
• We will study algorithms for solving discrete
  optimization problems

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Solution Process for discrete optimization
problems

• The role of Discrete Mathematics
• Show that the algorithm is correct
• Show that the algorithm is efficient
• Do careful mathematical analysis
• to design better algorithms

Optimiz. models
Abstraction, simplifications
Design algorithms
Solution to the model
Real life situation