Network flows and network design in theory and practice

Abstract

flows over time In this chapter, we study a generalization of network flows called abstract flows. This model replaces the underlying network structure by an abstract system of linearly ordered sets fulfilling a simple switching axiom. Abstract flows were introduced by Hoffman [Hof74] to investigate minimal structural requirements for obtaining max flow/min cut results. We extend his results by introducing a notion of time, generalizing Ford and Fulkerson’s concept of flows over time [FF58b]. Using the maximum abstract flow algorithm of McCormick [McC96], we show how maximum flows and minimum cuts over time can still be computed in the abstract setting. Publication remark: The results presented in this chapter are joint work with Jan-Philipp W. Kappmeier and Britta Peis [KMP12]. Ford and Fulkerson’s max flow/min cut theorem [FF56] is among the most influential results in combinatorial optimization. Understanding the driving forces behind this result is of great interest, not only for its fundamental importance to network flow theory itself but also due to the implied structural connection between cuts and connectivity in networks. Hoffman [Hof74] observed that the original proof of the theorem does not use the underlying network structure directly but only exploits one particular property of the path system, the so-called switching axiom: Whenever two paths P and Q intersect, there must be another path that is contained in the beginning of P and the end of Q. Hoffman succeeded in showing that a generalized version of the maximum flow problem defined on any set system fulfilling the switching axiom, called abstract network, still is totally dual integral (TDI). His structural results were later complemented by the combinatorial primal-dual algorithms of McCormick [McC96] and Martens and McCormick [MM08]. The high level of abstraction in Hoffman’s model leads to the question whether his results are restricted to the classic maximum flow problem or whether they extend to other variants of network flows. In this chapter, we introduce and investigate abstract flows over time and show how a temporally repeated abstract flow and a corresponding minimum cut can be computed by solving a single static weighted abstract flow problem— which in our case can be done even when accessing the abstract network through a very limited oracle. This immediately leads to the max flow/min cut theorem for abstract flows over time as our main result in this chapter. Although our construction resembles that of Ford and Fulkerson’s original result on (non-abstract) flows over time [FF58b], the proof turns out to be considerably more involved and we will need to take a detour via a relaxed version of abstract flows over time