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# Computing Vertex-Disjoint Paths in Large Graphs Using MAOs

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LIPIcs.ISAAC.2018.13.pdf
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## 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)
https://doi.org/10.4230/LIPIcs.ISAAC.2018.13

## 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
• Mathematics of computing → Graph algorithms
##### Keywords
• Computing Disjoint Paths
• Large Graphs
• Vertex-Connectivity
• Linear-Time
• Maximal Adjacency Ordering
• Maximum Cardinality Search
• Big Data
• Certifying Algorithm

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