Abstract
We consider the problem of computing $$k \in {\mathbb {N}}$$ 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 $$1 \le k \le \delta $$ (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 $$O(n+m)$$, 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.
Highlights
We extend our algorithm to compute several other path systems and discuss its impact for certifying algorithms
Vertex-connectivity is a fundamental parameter of graphs that, by a result due to Menger [12], can be characterized by the existence of internally vertex-disjoint paths between vertex pairs
Our research is driven by the question whether k internally vertex-disjoint paths can be computed faster deterministically. This question has particular impact for large graphs, as we aim for linear-time algorithms
Summary
Vertex-connectivity is a fundamental parameter of graphs that, by a result due to Menger [12], can be characterized by the existence of internally vertex-disjoint paths between vertex pairs. Our research is driven by the question whether k internally vertex-disjoint paths can be computed faster deterministically This question has particular impact for large graphs, as we aim for linear-time algorithms. All proofs known so far about vertex-connectivity in MAOs (including the ones by Henzinger and Nagamochi) are non-constructive and do not give any faster algorithm than the flow-based one for the initial question of computing internally vertex-disjoint paths. The perhaps most prominent such result is the minimum cut algorithm of Nagamochi and Ibaraki [14], which refines the work of Mader [10, 9], and was simplified by Frank [4] and by Stoer and Wagner [17] This algorithm computes iteratively a MAO and contracts the last two δ(-edge)-connected vertices of it. The 2-approximation algorithm for vertexconnectivity [6] by Henzinger can be made certifying using our new algorithm
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