Abstract
Given a directed bipartite graph \(G=(V,E)\) with a node set \(V=\{s\} \cup V_1 \cup V_2 \cup \{t\}\), and an arc set \(E=E_1\cup E_2\cup E_3\), where \(E_1=\{s\} \times V_1\), \(E_2= V_1 \times V_2\), and \(E_3=V_2 \times \{t\}\). Chen (1995) presented an \(O(|V| |E| \log (\frac{|V|^2}{|E|}))\) time algorithm to solve the parametric bipartite maximum flow problem. In this paper, we assume all arcs in \(E_2\) have infinite capacity [such a graph is called closure graph Hochbaum (1998)], and present a new approach to solve the problem, which runs in \(O(|V_1| |E| \log (\frac{|V_1|^2}{|E|}+2))\) time using Gallo et al’s parametric maximum flow algorithm, see Gallo et al. (1989). In unbalanced bipartite graphs, we have \(|V_1| << |V_2|\), so, our algorithm improves Chens’s algorithm in unbalanced and closure bipartite graphs.
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Ghiyasvand, M. Solving the parametric bipartite maximum flow problem in unbalanced and closure bipartite graphs. Ann Oper Res 229, 397–408 (2015). https://doi.org/10.1007/s10479-015-1807-7
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DOI: https://doi.org/10.1007/s10479-015-1807-7