The paper presents a series of algorithms for solving the Rooted Minor Containment problem and the Disjoint Paths problem in almost-linear fixed-parameter time.
The key ideas are:
For apex-minor-free graphs, the authors implement the irrelevant vertex technique efficiently using a dynamic treewidth data structure, allowing them to remove irrelevant vertices one by one in amortized no(1) time.
For clique-minor-free graphs, the authors reduce the problem to the apex-minor-free case by employing a version of the "recursive understanding" technique, which is implemented in almost-linear time using recent results on almost-linear time algorithms for flows and cuts.
The general case is reduced to the clique-minor-free case using a similar recursive scheme.
The authors also show that their results imply the existence of an n^(1+o(1))-time algorithm for deciding membership in every minor-closed class of graphs, as well as an Ok(m^(1+o(1)))-time algorithm for the Disjoint Paths problem.
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by Tuuk... klokken arxiv.org 04-08-2024
https://arxiv.org/pdf/2404.03958.pdfDypere Spørsmål