1 Discrete Optimization

2 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.

3 Example: making the scheduling model an optimization problem  There are 4 jobs that should be processed on the same machine. (Can’t 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. Job 3Job 1Job 4Job 2

4 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.

5 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

6 Ingredients of some common physical networks ApplicationPhysical 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 PipelinesWater, gas, oil

7 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

8 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 Real life situation Solution to the model Abstraction, simplifications Design algorithms