introduction-lp-duality1

The multicommodity ow problem is to determine if all

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Unformatted text preview: Alternative question: (easier, give the linear program) Explain what is doing each line of the program. Exercise 9.32 (Example for the Maximum Set Packing). Suppose you are at a convention of foreign ambassadors, each of which speaks English and other various languages. - French ambassador: French, Russian - US ambassador: - Brazilian ambassador: Portuguese, Spanish - Chinese ambassador: Chinese, Russian - Senegalese ambassador: Wolof, French, Spanish You want to make an announcement to a group of them, but because you do not trust them, you do not want them to be able to speak among themselves without you being able to understand them (you only speak English). To ensure this, you will choose a group such that no two ambassadors speak the same language, other than English. On the other hand you also want to give your announcement to as many ambassadors as possible. Write a linear program giving the maximum number of ambassadors at which you will be able to give the message. Exercise 9.33 (Maximum Set Packing (Dual of the set cover problem)). Given a finite set S and a list of subsets of S. Decision problem: Given an integer k, do there exist k pairwise disjoint sets (meaning, no two of them intersect)? Optimization problem: What is the maximum number of pairwise disjoint sets in the list? 9.4.5 Modelling Flow Networks and Shortest Paths. Definition 9.16 (Elementary flow network). A flow network is a four-tuple N = (D, s, t , c) where - D = (V, A) is a directed graph with vertice set V and arc set A. 156 CHAPTER 9. LINEAR PROGRAMMING - c is a capacity function from A to R+ ∪ ∞. For an arc a ∈ A, c(a) represents its capacity, that is the maximum amount of flow it can carry. - s and t are two distinct vertices: s is the source of the flow and t the sink. A flow is a function f from A to R+ which respects the flow conservation constraints and the capacity constraints. Exercise 9.34 (Maximum flow (Polynomial < Ford-Fulkerson)). Write the linear program solving the maximum flow problem for a flow network. Exercise 9.35 (Multicommodity flow). Consider a flow network N = (D, s, t , c). Consider a set of demands given by the matrix D = (di j ∈ R; i, j ∈ V, i ￿= j), where di j is the amount of flow that has to be sent from node i to node j. The multicommodity flow problem is to determine if all demands can be routed simultaneously on the network. This problem models a telecom network and is one of the fundamental problem of the networkin...
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