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A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities

Authors Julian Enoch, Kyle Fox, Dor Mesica, Shay Mozes

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ACM Subject Classification
  • Theory of computation → Network flows
Keywords
  • flow
  • planar graphs
  • vertex capacities

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Abstract

We give an O(k³ Δ n log n min(k, log² n) log²(nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks. This improves by a factor of max(k²,k log² n) over the fastest algorithm previously known for this problem [Wang, SODA 2019].
The speedup is obtained by two independent ideas. First we replace an iterative procedure of Wang that uses O(k) invocations of an O(k³ n log³ n)-time algorithm for maximum flow algorithm in a planar graph with k apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al. Second, we show two alternatives for computing flows in the k-apex graphs that arise in our modification of Wang’s procedure faster than the algorithm of Borradaile et al. In doing so, we introduce and analyze a sequential implementation of the parallel highest-distance push-relabel algorithm of Goldberg and Tarjan [JACM 1988].

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Julian Enoch, Kyle Fox, Dor Mesica, and Shay Mozes. A Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.72

Author Details

Julian Enoch
  • Department of Computer Science, University of Texas at Dallas, TX, USA
Kyle Fox
  • Department of Computer Science, University of Texas at Dallas, TX, USA
Dor Mesica
  • Efi Arazi School of Computer Science, The Interdisciplinary Center Herzliya, Israel
Shay Mozes
  • Efi Arazi School of Computer Science, The Interdisciplinary Center Herzliya, Israel

Funding

Enoch and Fox were partially supported by NSF grant CCF-1942597. Mesica and Mozes were partially supported by Israel Science Foundation grant No. 592/17.

References

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  2. Glencora Borradaile, Philip N. Klein, Shay Mozes, Yahav Nussbaum, and Christian Wulff-Nilsen. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. SIAM J. Comput., 46(4):1280-1303, 2017. URL: https://doi.org/10.1137/15M1042929.
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