AI 中文总结
本文针对一般无向平面图的最大st流问题，移除了现有近似算法的特殊限制，提出首个$\tilde O(D)$轮的分布式近最优$(1-o(1))$近似算法，核心为平面对偶图上的递归切口分布式实现。
AI 中文摘要
近年来，学界持续致力于为平面图的基础优化问题设计分布式算法。其中，单源最短路径问题存在定向平面图上的$\tilde O(D^2)$轮精确算法[Li, Parter STOC'19]，以及无向平面图上的$\tilde O(D)$轮$(1+o(1))$近似算法[Rozhon, Grunau, Haeupler, Zuzic, Li STOC'22]（$D$为图的跳直径）。近期[Abd-Elhaleem, Dory, Parter, Weimann PODC'25]针对最大st流问题，得到了定向平面图上匹配的$\tilde O(D^2)$轮精确算法，但近似情形下仅给出无向平面图上$D \times n^{o(1)}$轮的$(1-o(1))$近似算法，且仅适用于$s$和$t$位于同一面的特殊情况。本文移除了$s$和$t$需位于同一面的限制（同时消除了$n^{o(1)}$因子），首次提出一般无向平面图上最大st流问题的分布式近最优$\tilde O(D)$轮$(1-o(1))$近似算法。核心技术贡献是对经典Reif[SICOMP'83]集中式算法的分布式实现，通过对图$G$的平面对偶图$G^*$执行精细的递归切口过程完成，难点在于需在仅能通过输入图$G$通信的条件下，模拟对偶图$G^*$上的动态变化（切口操作）。
英文摘要
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 $\tilde O(D^2)$-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 $\tilde O(D^2)$-rounds exact algorithm for directed planar graphs. However, for the approximate case, they give a $D\cdot n^{o(1)}$-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 $\tilde{O}(D)$-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 $G^*$ 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 $G$.
Comments62 pages. 25 figures. A shortened version is to appear in DISC 2026
