Abstract
Processing networks (cf. Koene in Minimal cost flow in processing networks: a primal approach, 1982) and manufacturing networks (cf. Fang and Qi in Optim Methods Softw 18:143–165, 2003) are well-studied extensions of traditional network flow problems that allow to model the decomposition or distillation of products in a manufacturing process. In these models, so called flow ratios \(\alpha _e \in [0,1]\) are assigned to all outgoing edges of special processing nodes. For each such special node, these flow ratios, which are required to sum up to one, determine the fraction of the total outgoing flow that flows through the respective edges. In this paper, we generalize processing networks to the case that these flow ratios only impose an upper bound on the respective fractions and, in particular, may sum up to more than one at each node. 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 show that there exists a fully polynomial-time approximation scheme (FPTAS) for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic and we provide two exact algorithms with strongly polynomial running-time for the problem on series–parallel graphs. Finally, we study the case of integral flows and show that the problem becomes \({\mathcal {NP}}\)-hard to solve and approximate in this case.
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Notes
Note that, by setting \(\alpha _e = 1\) for all \(e \in \delta ^+(v)\), we can model nodes v without any restrictions on the flow ratios (often called transshipment nodes in the context of processing networks). Hence, our model generalizes both the maximum flow problem in traditional networks and in processing networks.
A feasible circulation x is a feasible flow that fulfills \({{\mathrm{excess}}}_x(v) = 0\) for each node \(v \in V\).
Clearly, this does not imply that the values \(\alpha _e\) of all outgoing edges of some node v sum up to at most one, in contrast to the case of traditional processing networks.
An s–t-cut (S, T) is a partition of the node set into two disjoint sets S and T with \(s \in S\) and \(t \in T\) (cf. Ahuja et al. 1993). In the following, we let \(\delta ^+(X)\) (\(\delta ^-(X)\)) for \(X \subset V\) denote the set of edges \(e=(v,w)\) with \(v \in X\) and \(w \notin X\) (\(v \notin X\) and \(w \in X\)). We then also identify the s–t-cut (S, T) with the set \(\delta ^+(S)\) of edges in the cut, i.e., crossing the cut in forward direction.
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Acknowledgments
We thank the anonymous referees for their valuable comments and suggestions that helped to improve the presentation of the paper. This work was partially supported by the German Federal Ministry of Education and Research within the project “SinOptiKom – Cross-sectoral Optimization of Transformation Processes in Municipal Infrastructures in Rural Areas”.
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Holzhauser, M., Krumke, S.O. & Thielen, C. Maximum flows in generalized processing networks. J Comb Optim 33, 1226–1256 (2017). https://doi.org/10.1007/s10878-016-0031-y
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DOI: https://doi.org/10.1007/s10878-016-0031-y