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# Computing vertex-disjoint paths in large graphs using MAOS

Publikationstyp

Conference Paper

Publikationsdatum

2018-12-06

Sprache

English

TORE-URI

First published in

Number in series

123

Article Number

13

Citation

29th International Symposium on Algorithms and Computation (ISAAC 2018) 123: 13 (2018)

Contribution to Conference

Publisher DOI

Scopus ID

Publisher

Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH, Dagstuhl Publishing

We consider the problem of computing k ∈ N internally vertex-disjoint paths between special vertex pairs of simple connected graphs. For general vertex pairs, the best deterministic time bound is, since 42 years, O(min{k, n}m) for each pair by using traditional flow-based methods. The restriction of our vertex pairs comes from the machinery of maximal adjacency orderings (MAOs). Henzinger showed for every MAO and every 1 ≤ k ≤ δ (where δ is the minimum degree of the graph) the existence of k internally vertex-disjoint paths between every pair of the last δ − k + 2 vertices of this MAO. Later, Nagamochi generalized this result by using the machinery of mixed connectivity. Both results are however inherently non-constructive. We present the first algorithm that computes these k internally vertex-disjoint paths in linear time O(m), which improves the previously best time O(min{k, n}m). Due to the linear running time, this algorithm is suitable for large graphs. The algorithm is simple, works directly on the MAO structure, and completes a long history of purely existential proofs with a constructive method. We extend our algorithm to compute several other path systems and discuss its impact for certifying algorithms.