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Õptimal Distributed Maximum Flow Approximation in Undirected Planar Graphs

Authors Yaseen Abd-Elhaleem , Michal Dory , Oren Weimann

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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Maximum flow
  • minimum st-cut
  • shortest paths
  • planar graphs
  • distributed computing

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Abstract

Persistent efforts in recent years have been devoted to devising distributed algorithms for fundamental optimization problems in planar graphs. In particular, for Single-Source Shortest-Paths, there is an Õ(D²)-rounds exact algorithm [Li, Parter STOC'19] for directed planar graphs, and an Õ(D)-rounds (1+o(1))-approximation algorithm [Rozhon, Grunau, Haeupler, Zuzic, Li STOC'22] for undirected planar graphs (where D is the graph’s hop-diameter). Recently [Abd-Elhaleem, Dory, Parter, Weimann PODC'25], a matching bound for the exact case was obtained for the Maximum st-Flow problem. Namely, an Õ(D²)-rounds exact algorithm for directed planar graphs. However, for the approximate case, they give a D⋅ n^o(1)-rounds (1-o(1))-approximation algorithm for undirected planar graphs that works only for the special case where both s and t lie on the same face. 
In this paper, we remove the restriction that both s and t must lie on the same face (we also eliminate the n^o(1) factor). Namely, we present the first distributed near-optimal Õ(D)-rounds (1-o(1))-approximation algorithm for Maximum st-Flow in general undirected planar graphs. Our main technical contribution is a distributed implementation of the classical Reif’s [SICOMP'83] centralized algorithm. This is achieved by a careful recursive incision procedure on the planar dual G^* of the graph G. It is challenging, because we need to simulate dynamic changes (incisions) over the dual graph G^*, while we can only communicate over the input graph G.

Cite As Get BibTex

Yaseen Abd-Elhaleem, Michal Dory, and Oren Weimann. Õptimal Distributed Maximum Flow Approximation in Undirected Planar Graphs. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.19

Author Details

Yaseen Abd-Elhaleem
  • Department of Computer Science, University of Haifa, Israel
Michal Dory
  • Department of Computer Science, University of Haifa, Israel
Oren Weimann
  • Department of Computer Science, University of Haifa, Israel

Funding

  • Abd-Elhaleem, Yaseen: supported in part by the Israel Science Foundation grants No. 810/21 and 2829/25.
  • Dory, Michal: supported in part by the Israel Science Foundation grant No. 2829/25.
  • Weimann, Oren: supported in part by the Israel Science Foundation grant No. 365/26.

References

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