duality - Chapter 4 Introduction to duality Suppose your...

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Chapter 4 Introduction to duality Suppose your boss asked you to solve an optimization problem and that you were able to fnd a Feasible solution x For it. Your boss now naturally asks you “Is x optimal?” Suppose that you know the answer to the question, maybe because you’re a genius, or maybe because you worked For days on this problem. But in the end, you will have to convince your boss that your answer is correct, and he is not a genius and too busy to work For days on the problem. IF the answer is “no”, there is an easy way to do this – just show your boss a better solution. IF the answer is “yes”, we want a similarly easy way to convince your boss that every Feasible solution has objective value at most that oF x . In the previous chapter, we have shown that when the simplex algorithm fnds an optimal basic solution x it also provides a prooF that no solution has value larger than that oF x . In this chapter, we introduce the concept oF duality which allows us to generate succinct certifcates oF optimality For optimal solutions to linear programs. We can also obtain such certifcates oF optimality For special types oF integer programs . Duality, as we will see, is however even more powerFul than that, and can be used in the design oF algorithms to solve optimization problems. 81
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82 CHAPTER 4. INTRODUCTION TO DUALITY 4.1 A frst example: Shortest paths Let us start with an example. Recall the shortest path problem from Chapter 1, where we are given a graph G =( V , E ) , non-negative lengths c e for all edges e 2 E , and two speciFc vertices s , t 2 V . An st -path P is a sequence of edges v 1 v 2 , v 2 v 3 ,..., v k - 2 v k - 1 , v k - 1 v k in E such that v 1 = s , v k = t , and v i 6 = v j for all i 6 = j . We are looking for an -path of minimum total length  ( c e : e 2 P ) . 3 2 1 2 2 4 1 4 s t a d c b The Fgure on the right shows an instance of this problem. Each of the edges in the graph is labeled by its length. The thick black edges in the graph form an -path P of total length 7. Is this a shortest path? The answer is “yes”, but how could you convince your boss? 3 2 1 2 2 4 1 4 s t a d c b ({s},3) ({s,a,b,c,d},1) ({s,a,c},2) ({s,a},1) Here is a nice way of accomplish- ing this. The Fgure on the left shows the same graph as before together with a set of four moats each of which sep- arates s from t . Each moat is labeled by a pair ( S i , y i ) , where 1. S i is the set of vertices of V that are on the s -side of the moat, and 2. y i is the width of the moat. We say that an edge uv 2 E crosses a moat S i if u is in S i , and v is outside, in V \ S i . ±or example, edge ab in the Fgure on the left crosses moats { s , a } and { s , a , c } . We say that a collection S = { ( S 1 , y 1 ) ( S q , y q ) } of moats and their widths is feasible if (a) each moat separates s from t ; i.e., s 2 S i and t 2 V \ S i for all i , and
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4.2. A SECOND EXAMPLE: MAXIMUM WEIGHT MATCHING 83 (b) if each edge crosses moats of total width no larger than its length; i.e., Â ( y i : u 2 S i , v 2 V \ S i ) c uv , for all uv 2 E .
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This note was uploaded on 11/15/2011 for the course CO 227 taught by Professor 1 during the Fall '10 term at Waterloo.

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duality - Chapter 4 Introduction to duality Suppose your...

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