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Unformatted text preview: Integer Programming IE418 Lecture 5 Dr. Ted Ralphs IE418 Lecture 5 1 Reading for This Lecture Wolsey Chapter 6 N&W Sections I.5.1 and I.5.2 IE418 Lecture 5 2 Introduction to Computational Complexity What is the goal of computational complexity theory? To provide a method of quantifying problem difficulty in an absolute sense. To provide a method comparing the relative difficulty of two different problems. We would like to be able to rigorously define the meaning of efficient algorithm . Complexity theory is built on a basic set assumptions called the model of computation . We will not concern ourselves too much with the details of a particular model here. To deal with this topic in full rigor would require a full semester course. IE418 Lecture 5 3 Problems and Instances What is the difference between a problem and a problem instance ? To define these terms rigorously takes a great deal of mathematical machinery. We will do so only within the context of mathematical programming. Loosely, a problem or model is an infinite family of instances whose objective function and constraints have a specific structure. An instance is obtained by specifying values for the various problem parameters. Recall the distinction between model and data in AMPL. IE418 Lecture 5 4 Measuring the Difficulty of an Instance Loosely speaking, the difficulty of a problem instance is easy to judge. Try to solve the problem instance and see how long it takes ( the running time ). Note that this inherently depends on the algorithm and the computing platform . We want a measure independent of both these variables. We will always assume the best known algorithm and best known implementation are used. We will measure execution time in terms of the total number of elementary operations executed (more on this later). IE418 Lecture 5 5 Measuring the Difficulty of a Problem On the previous slide, we discussed how to measure the difficulty of an instance. The difficulty of a problem is harder to define. Possible methods of evaluation Best case running time Average case running time Worst case running time Best case doesnt given us any guarantee about the difficulty of a given instance. Average case is difficult to analyze and depends on specifying a probability distribution on the instances....
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This note was uploaded on 08/06/2008 for the course IE 418 taught by Professor Ralphs during the Spring '08 term at Lehigh University .
 Spring '08
 Ralphs

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