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Unformatted text preview: CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (saberi@stanford.edu) February 4, 2010 Lecture 10: Linear Programs Maximum flow and many other problems can be incorporated into a broader class of problems called linear programs (LPs) . There are several courses at Stanford that deal specifically with LPs (such as MS&E 310). The simplex method , which is the algorithm for solving LP, has been called as one of the most important algorithms of the 20 th century. 1 The General, the Cook, and the Mother of all MaxMin Theorems Suppose we have a division of soldiers, a General, and a Cook. There is a set of food ingredients { 1 , 2 ,...,n } , where one unit of i provides a ij units of property j , 1 j m (such as calories, protein, vitamins, etc...). The cook needs to ensure that each soldier gets a sufficient amount of property j which is given by b j ; in other words, the nutrition fact of the food should be such that each soldier gets at least b j units of property j ; consuming one unit of i costs c i . The general asks the cook to prepare a meal (regardless of taste) that meets all of the soldiers requirements at a minimum cost. Let x i be amount of ingredient i consumed in the food, the problem that the cook needs to solve can be formulated as follows: minimize: X i c i x i subject to: X i a ij x i b j for all j x i for all i Such an optimization problem is called a linear program (LP) as the constraints and the objective function are both linear. Example: The list of ingredients and their nutrition facts are given in Table 1; each soldier needs 6 units of vitamins, 9 units of calories, and 5 units of proteins. Ingredient Vitamins Calories Proteins Cost Milk 3 2 2 8 Potato 3 3 Meat 2 1 2 Fruit 1 1 1 3 Table 1: Ingredients and Their Nutrition Facts It is easy to check that 2 units of milk, 1 unit of potato, and 1 unit of meat satisfy all the constraints, and cost 2 8 + 3 + 2 = 21. How can the cook show that this is, in fact, the recipe with the minimum cost?...
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