Arnold Filtser, Nikolai Karol, Hung Le, Florian Lehner
原文摘要
Given an $H$-induced-minor-free graph $G$ with $n$ vertices, we construct a balanced separator of $G$ consisting of $\tilde{O}(|V(H)|\sqrt{n})$ star subgraphs. Our result is an induced-minor-free analog of the famous separator theorem for $H$-minor-free graphs by Alon, Seymour, and Thomas [J. Amer. Math. Soc., 1990]. A key idea in our construction is a new decomposition of $H$-induced-minor-free graphs, analogous to the cop decomposition for $H$-minor-free graphs by Andreae [J. Combin. Theory Ser. B, 1986]. We also discuss several algorithmic applications of our result.
Arnold Filtser, Gramoz Goranci, Ali Momeni, Neel Patel, Maximilian Probst Gutenberg
原文摘要
We study the fully-dynamic all-pairs minimum cut problem on planar graphs. Given an $n$-vertex undirected planar graph $G=(V,E)$ undergoing edge insertions and deletions, the goal is to efficiently process these updates while supporting $s$-$t$ minimum cut queries for any pair of vertices $s$ and $t$. We give a $(1+\varepsilon)$-approximate dynamic algorithm that supports edge updates and minimum cut queries in $n^{o(1)}$ time, for any $1/\text{poly}(\log n) < \varepsilon < 1$. For general graphs, previous results either are limited to partially dynamic settings or achieve only logarithmic-factor approximations. The main conceptual and technical contribution is to show that a suitable adaptation of the planar emulator of Chang, Krauthgamer, and Tan~[STOC'22] satisfies the stronger pattern emulator introduced by Chen and Tan~[ICALP'25]. This leads almost immediately to a nearly linear-time construction of pattern emulators, which, when combined with the recent fully dynamic $(1+\varepsilon)$-approximate all-pairs shortest paths algorithm of Filtser, Goranci, Patel, and Probst Gutenberg~[FOCS'24], yields our result.