Just Accepted
Flow-augmentation I: Directed graphs
Abstract
We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph G, two vertices s, t ∈ V(G), and an integer k, adds (randomly) to G a number of arcs such that for every minimal st-cut Z in G of size at most k, with probability 2− poly(k) the set Z becomes a minimum st-cut in the resulting graph. We also provide a deterministic counterpart of this procedure.
The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set whose parameterized complexity status was repeatedly posed as open problems:By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph H, if the List H-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable.
(1)
Chain SAT, defined by Chitnis, Egri, and Marx [ESA’13, Algorithmica’17],
(2)
a number of weighted variants of classic directed cut problems, such as Weighted st-Cut or Weighted Directed Feedback Vertex Set.
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[16]
Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. 2024. Flow-augmentation II: Undirected graphs. ACM Trans. Algorithms (jan 2024). https://doi.org/10.1145/3641105 Just Accepted.
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Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. 2024. Flow-augmentation II: Undirected Graphs. ACM Trans. Algorithms 20, 2, Article 12 (mar 2024), 26 pages. https://doi.org/10.1145/3641105
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Index Terms
- Flow-augmentation I: Directed graphs
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Publication History
Online AM: 30 November 2024
Accepted: 06 November 2024
Revised: 25 July 2024
Received: 16 February 2023
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