Abstract
Up to now, we have considered ows or circulations only on a given network. But it is also quite interesting to study the reverse problem of designing a network—as economically as possible—on which a ow meeting given requirements can be realized. In practical terms, one might think of planning a system of roads. In the present chapter, we shall mainly study two types of network design problems. On the one hand, we consider the case where all edges may be built with the same cost, and where we are looking for an undirected network with lower bounds on the maximal values of the ows between any two vertices. Both the analysis and design of such symmetric networks use so-called equivalent ow trees; this technique also has an interesting application for the construction of certain communication networks which will be the topic of Sect. 11.4. On the other hand, we shall address the question of how one may increase the maximal value of the ow for a given ow network by increasing the capacities of some edges by the smallest possible amount.
What thought and care to determine the exact site for a bridge, or for a fountain, and to give a mountain road that perfect curve which is at the same time the shortest…
Marguerite Yourcenar
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Notes
- 1.
Using this rather sloppy notation (that is, using the same symbol w for the weight function on T as well as for the flow function on N) is justified, as w(x,y)=w(e) for each edge e=xy of T.
- 2.
Actually u(x)=u′(x) for all x, but we do not need this for our proof.
- 3.
If desired, we may avoid parallel edges by subdividing e′.
- 4.
Vertices in S can be reached from s by a path of cost 0, so that we want to direct our path from S to T, not reversely.
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Jungnickel, D. (2013). Synthesis of Networks. In: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32278-5_12
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