2026-07-22T20:50:16Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9961 2024-03-06T10:45:12Z ddc:004 open_access

Computing Vertex-Disjoint Paths in Large Graphs Using MAOs Preißer, Johanna E. Schmidt, Jens M. Computing Disjoint Paths Large Graphs Vertex-Connectivity Linear-Time Maximal Adjacency Ordering Maximum Cardinality Search Big Data Certifying Algorithm 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Johanna E. Preißer and Jens M. Schmidt 2018 Is Part Of LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2018.13 urn:nbn:de:0030-drops-99613 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.13 eng https://creativecommons.org/licenses/by/3.0/legalcode