Almost-Linear Time Parameterized Algorithm for Rankwidth via Dynamic Rankwidth
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| Titel: | Almost-Linear Time Parameterized Algorithm for Rankwidth via Dynamic Rankwidth |
|---|---|
| Autoren: | Tuukka Korhonen, Marek Sokołowski |
| Quelle: | Proceedings of the 56th Annual ACM Symposium on Theory of Computing. :1538-1549 |
| Publication Status: | Preprint |
| Verlagsinformationen: | ACM, 2024. |
| Publikationsjahr: | 2024 |
| Schlagwörter: | Computer Science - Data Structures and Algorithms, Mathematics - Combinatorics, Computer Science - Discrete Mathematics |
| Beschreibung: | We give an algorithm that given a graph $G$ with $n$ vertices and $m$ edges and an integer $k$, in time $O_k(n^{1+o(1)}) + O(m)$ either outputs a rank decomposition of $G$ of width at most $k$ or determines that the rankwidth of $G$ is larger than $k$; the $O_k(\cdot)$-notation hides factors depending on $k$. Our algorithm returns also a $(2^{k+1}-1)$-expression for cliquewidth, yielding a $(2^{k+1}-1)$-approximation algorithm for cliquewidth with the same running time. This improves upon the $O_k(n^2)$ time algorithm of Fomin and Korhonen [STOC 2022]. The main ingredient of our algorithm is a fully dynamic algorithm for maintaining rank decompositions of bounded width: We give a data structure that for a dynamic $n$-vertex graph $G$ that is updated by edge insertions and deletions maintains a rank decomposition of $G$ of width at most $4k$ under the promise that the rankwidth of $G$ never grows above $k$. The amortized running time of each update is $O_k(2^{\sqrt{\log n} \log \log n})$. The data structure furthermore can maintain whether $G$ satisfies some fixed ${\sf CMSO}_1$ property within the same running time. We also give a framework for performing ``dense'' edge updates inside a given set of vertices $X$, where the new edges inside $X$ are described by a given ${\sf CMSO}_1$ sentence and vertex labels, in amortized $O_k(|X| \cdot 2^{\sqrt{\log n} \log \log n})$ time. Our dynamic algorithm generalizes the dynamic treewidth algorithm of Korhonen, Majewski, Nadara, Pilipczuk, and Soko{\l}owski [FOCS 2023]. |
| Publikationsart: | Article |
| DOI: | 10.1145/3618260.3649732 |
| Zugangs-URL: | http://arxiv.org/abs/2402.12364 |
| Rights: | URL: https://www.acm.org/publications/policies/copyright_policy#Background |
| Dokumentencode: | edsair.doi.dedup.....ef01de68d9aada2f52685f443d19e10c |
| Datenbank: | OpenAIRE |
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