Abstract
FlowExpansion is a network design problem, in which the input consists of a flow network and a set of candidate edges, which may be added to the network. Adding a candidate incurs given costs. The goal is to determine the cheapest set of candidate edges that, if added, allow the demands to be satisfied. FlowExpansion is a variant of the Minimum-Cost Flow problem with non-linear edge costs.
We study FlowExpansion for both graph-theoretical and electrical flow networks. In the latter case this problem is also known as the Transmission Network Expansion Planning problem. We give a structured view over the complexity of the variants of FlowExpansion that arise from restricting, e.g., the graph classes, the capacities, or the number of sources and sinks. Our goal is to determine which restrictions have a crucial impact on the computational complexity. The results in this paper range from polynomial-time algorithms for the more restricted variants over \(\mathcal {NP}\)-hardness proofs to proofs that certain variants are \(\mathcal {NP}\)-hard to approximate even within a logarithmic factor of the optimal solution.
This work was funded (in part) by the Helmholtz Program Storage and Cross-linked Infrastructures, Topic 6 Superconductivity, Networks and System Integration and by the German Research Foundation (DFG) as part of the Research Training Group GRK 2153: Energy Status Data – Informatics Methods for its Collection, Analysis and Exploitation.
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Wagner, D., Wolf, M. (2021). The Complexity of Flow Expansion and Electrical Flow Expansion. In: Bureš, T., et al. SOFSEM 2021: Theory and Practice of Computer Science. SOFSEM 2021. Lecture Notes in Computer Science(), vol 12607. Springer, Cham. https://doi.org/10.1007/978-3-030-67731-2_32
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DOI: https://doi.org/10.1007/978-3-030-67731-2_32
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