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Computing vertex-disjoint paths in large graphs using MAOS
Publikationstyp
Conference Paper
Date Issued
2018-12-06
Sprache
English
Author(s)
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
ISBN of container
978-395977094-1
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.
Subjects
Big Data
Certifying Algorithm
Computing Disjoint Paths
Large Graphs
Linear-Time
Maximal Adjacency Ordering
Maximum Cardinality Search
Vertex-Connectivity