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Computing Vertex-Disjoint Paths in Large Graphs Using MAOs

Authors: Johanna E. Preißer and Jens M. Schmidt

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)


Abstract
We consider the problem of computing k in 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,sqrt{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 <= delta (where delta is the minimum degree of the graph) the existence of k internally vertex-disjoint paths between every pair of the last delta-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,sqrt{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.

Cite as

Johanna E. Preißer and Jens M. Schmidt. Computing Vertex-Disjoint Paths in Large Graphs Using MAOs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 13:1-13:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{preier_et_al:LIPIcs.ISAAC.2018.13,
  author =	{Prei{\ss}er, Johanna E. and Schmidt, Jens M.},
  title =	{{Computing Vertex-Disjoint Paths in Large Graphs Using MAOs}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{13:1--13:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.13},
  URN =		{urn:nbn:de:0030-drops-99613},
  doi =		{10.4230/LIPIcs.ISAAC.2018.13},
  annote =	{Keywords: Computing Disjoint Paths, Large Graphs, Vertex-Connectivity, Linear-Time, Maximal Adjacency Ordering, Maximum Cardinality Search, Big Data, Certifying Algorithm}
}
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