Generalized Processing Networks

Abstract

We turn our considerations to a generalization of the maximum flow problem in which each edge $e = \left( {v,w} \right) \in \,\text{E}$ is assigned with a so called flow ratio $\alpha_{e} \in \left[ {0,\,1} \right]$ that imposes an upper bound on the fraction of the total outgoing flow at v that may be routed through the edge e. This model embodies a generalization of the maximum flow problem in processing networks (Koene, 1982), in which the corresponding flow ratios specify the exact fraction of flow rather than only an upper bound. We show that a flow decomposition similar to the one for traditional network flows is possible and can be computed in strongly polynomial time. Moreover, we prove that the problem is at least as hard to solve as any packing LP but that there also exists a fully polynomialtime approximation scheme for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic. For the case of series-parallel graphs, we provide two exact algorithms with strongly polynomial running time. Finally, we study the case of integral flows and show that the problem becomes ${\mathcal{N}\mathcal{P}}$-hard to solve and approximate in this case.

This chapter is based on joint work with Sven O. Krumke and Clemens Thielen (Holzhauser et al., 2016c).