arXiv:2611.05756 · v1

Flow-Optimal Separators for Minor-Free and Induced-Minor-Free Graphs in Linear Time via Follow the Regularized Leader

首次提交 2026-11-04 07:55

版本更新 2026-11-04 07:55

雷达观测 2026-11-05 10:08

摘要

We give a new algorithm for finding balanced separators in minor-free and induced-minor-free graphs. A key technical idea is a new connection to online learning: we formulate a continuous, non-convex optimization problem and solve it using the Follow the Regularized Leader (FTRL) framework. Our implementation of each FTRL step yields a purely combinatorial algorithm that assigns and adjusts vertex weights over multiple rounds. For graphs with $n$ vertices and $m$ edges, we obtain the following results. (1) For $H$-induced-minor-free graphs, we find a balanced separator of size $O(\log(|V(H)|)\sqrt{|V(H)|+|E(H)|}\sqrt{m})$ in expected linear time for fixed $H$. As a corollary, we find balanced separators of size $O(\sqrt{m})$ in string graphs in expected linear time, without requiring string representations. (2) For $K_h$-minor-free graphs, we find a balanced separator of size $O(h\log h\sqrt{n})$ in expected linear time for fixed $h$. Our algorithm unifies two long lines of work on separators in minor-free graphs, dating back to Alon, Seymour, and Thomas (1990): one optimizing separator size and the other optimizing running time. It also gives a natural interpretation of the vertex weighting scheme in a recent linear-time algorithm by Bonnet, Korhonen, Le, Li, and Masarik (2026). The separator bounds for both graph classes are flow-optimal: techniques based on flow-cut duality, including ours, cannot yield balanced separators of asymptotically smaller size.

作者备注：34 pages, 4 figures