Minor Containment and Disjoint Paths in Almost-Linear Time
We give an algorithm that, given graphs G and H , tests whether H is a minor of G in time \mathcal{O}_{H}(\overline{n}^{1+o(1)}) ; here, n is the number of vertices of G and the \mathrm{O}_{H}(.) -notation hides factors that depend on H and are computable. By the Graph Minor Theorem, this implies th...
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| Vydáno v: | Proceedings / annual Symposium on Foundations of Computer Science s. 53 - 61 |
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| Hlavní autoři: | , , |
| Médium: | Konferenční příspěvek |
| Jazyk: | angličtina |
| Vydáno: |
IEEE
27.10.2024
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| Témata: | |
| ISSN: | 2575-8454 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We give an algorithm that, given graphs G and H , tests whether H is a minor of G in time \mathcal{O}_{H}(\overline{n}^{1+o(1)}) ; here, n is the number of vertices of G and the \mathrm{O}_{H}(.) -notation hides factors that depend on H and are computable. By the Graph Minor Theorem, this implies the existence of an n^{1+o(1)} -time membership test for every minor-closed class of graphs. More generally, we give an \mathcal{O}_{H,\vert X\vert} (m^{1+o(1)}) -time algorithm for the rooted version of the problem, in which G comes with a set of roots X\subseteq V(G) and some of the branch sets of the sought minor model of H are required to contain prescribed subsets of X ; here, m is the total number of vertices and edges of G . This captures the Disjoint Pathsproblem, for which we obtain an \mathcal{O}_{k}(m^{1+o(1)\backslash } -time algorithm, where k is the number of terminal pairs. For all the mentioned problems, the fastest algorithms known before are due to Kawarabayashi, Kobayashi, and Reed [JCTB 2012], and have a time complexity that is quadratic in the number of vertices of G . Our algorithm has two main ingredients: First, we show that by using the dynamic treewidth data structure of Korhonen, Majewski, Nadara, Pilipczuk, and Sokolowski [FOCS 2023], the irrelevant vertex technique of Robertson and Seymour can be implemented in almost-linear time on apex-minor-free graphs. Then, we apply the recent advances in almost-linear time flow/cut algorithms to give an almost-linear time implementation of the recursive understanding technique, which effectively reduces the problem to apex-minor-free graphs. |
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| ISSN: | 2575-8454 |
| DOI: | 10.1109/FOCS61266.2024.00014 |