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.
@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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