Discrete_Opt - DISCRETE OPTIMIZATION Notes prepared for:...

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DISCRETE OPTIMIZATION Notes prepared for: MATH 2602 Linear and Discrete Mathematics Fall 1999 1
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Table of Contents 1. Introduction to Some Discrete Problems ................. 1 1.1 Illustrations ......................................... 1 1.2 Overview ............................................ 4 2. Formulations 5 3. Solutions ................................................ 8 3.1 Branch-and-Bound ................................... 9 3.1.1 Branch-and-Bound Solution to the Knapsack Problem ............................... 11 3.1.2 Branch-and-Bound Solution to the Traveling Salesman Problem ...................... 14 3.2 Direct Approaches .................................. 20 3.2.1 1-Machine Scheduling ............................. 21 3.2.2 Vertex Covering on Trees ......................... 22 3.2.3 Matching 25 4. Chinese Postman Problem .............................. 30 5. Approximation Procedures 34 5.1 Bin-Packing ........................................ 35 5.2 Vertex Cover ....................................... 37 5.3 Traveling Salesman Problem ........................ 39 6. Final Comments 42 7. Exercises ............................................... 43 8. References ............................................. 45 2
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DISCRETE OPTIMIZATION Earlier, we were introduced to the rudiments of linear programming (LP), a methodology, which to this day remains an exceptionally powerful tool in the problem-solving arsenal of mathematicians, engineers, systems planners, and other analysts. With the ever-increasing power of modern day comput- ers, commercial LP software is now routinely available for the solution of truly large linear optimization models. Unfortunately, however, there are many problems that arise in practice, and hence, are of substantial impor- tance, but that are simply not amenable to modeling and solution as linear programs. This state of affairs suggests, of course, that analysts have to enhance their array of tools in order to effectively deal with these problems, many of which are exceptionally difficult. In this document, we examine a small sample of the sorts of the problems that reside beyond the realm of routine LP-solvability and we look at some of the strategies that are possible in dealing with them. 1 Introduction to Some Discrete Problems We begin by considering a few real-world examples. For the sake of brevity, the presentation of some of these may seem overly simplistic and/or con- trived; however, in each case there are many manifestations of the problem setting described that occur every day in the world of practice. As a conse- quence, it is essential that effective strategies for their solution be found. 1.1 Illustrations Example 1 . Within a large geographical region there are a number of localities where television transmitters can be located. The cost of locating transmitters can be substantial and varies with locales. Find a placement of transmitters in order that all customers in the region are served and do so at minimum total location cost. Example 2 . During the Second World War, some allied air squadrons uti- lized planes requiring two pilots. However, not all pairings of pilots were admissable due to differences in language and/or technical expertise. During 1
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combat runs, the aim was to have as many planes as possible in the air at once. How could this be easily guaranteed?
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This note was uploaded on 11/08/2009 for the course MATH 2602 taught by Professor Costello during the Spring '09 term at Georgia Tech.

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Discrete_Opt - DISCRETE OPTIMIZATION Notes prepared for:...

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