Solving the parametric bipartite maximum flow problem in unbalanced and closure bipartite graphs

Mehdi Ghiyasvand ()

Annals of Operations Research, 2015, vol. 229, issue 1, 397-408

Abstract: Given a directed bipartite graph $$G=(V,E)$$ G = ( V , E ) with a node set $$V=\{s\} \cup V_1 \cup V_2 \cup \{t\}$$ V = { s } ∪ V 1 ∪ V 2 ∪ { t } , and an arc set $$E=E_1\cup E_2\cup E_3$$ E = E 1 ∪ E 2 ∪ E 3 , where $$E_1=\{s\} \times V_1$$ E 1 = { s } × V 1 , $$E_2= V_1 \times V_2$$ E 2 = V 1 × V 2 , and $$E_3=V_2 \times \{t\}$$ E 3 = V 2 × { t } . Chen ( 1995 ) presented an $$O(|V| |E| \log (\frac{|V|^2}{|E|}))$$ O ( | V | | E | log ( | V | 2 | E | ) ) time algorithm to solve the parametric bipartite maximum flow problem. In this paper, we assume all arcs in $$E_2$$ 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 $$O(|V_1| |E| \log (\frac{|V_1|^2}{|E|}+2))$$ O ( | V 1 | | E | log ( | 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 $$|V_1| >> |V_2|$$ | V 1 | > > | V 2 | , so, our algorithm improves Chens’s algorithm in unbalanced and closure bipartite graphs. Copyright Springer Science+Business Media New York 2015

Keywords: Network flows; The parametric bipartite maximum flow problem; Closure graphs; Unbalanced bipartite graphs; Gallo et al’s parametric maximum flow algorithm (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10479-015-1807-7

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