$\tilde{\text{O}}$ptimal Distributed Maximum Flow Approximation in Undirected Planar Graphs

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 $\tilde O(D<sup>2)$-rounds exact algorithm [Li, Parter STOC'19] for directed planar graphs, and an $\tilde {O}(D)$-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 $\tilde O(D<sup>2)$-rounds exact algorithm for directed planar graphs. However, for the approximate case, they give a $D\cdot n<sup>{o(1)}$-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 $n<sup>{o(1)}$ factor). Namely, we present the first distributed near-optimal $\tilde{O}(D)$-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 $G<sup>*$ of the graph $G$. It is challenging, because we need to simulate dynamic changes (incisions) over the dual graph $G<sup>*$, while we can only communicate over the input graph $G$.