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Unformatted text preview: Alternative question: (easier, give the linear program) Explain what is doing each line of the
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)
- 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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- Fall '13
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