
A relaxation of the Directed Disjoint Paths problem: a global congestion metric helps
In the Directed Disjoint Paths problem, we are given a digraph D and a s...
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Faster parameterized algorithm for pumpkin vertex deletion set
A directed graph G is called a pumpkin if G is a union of induced paths ...
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Efficient Graph Minors Theory and Parameterized Algorithms for (Planar) Disjoint Paths
In the Disjoint Paths problem, the input consists of an nvertex graph G...
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The Power of CutBased Parameters for Computing Edge Disjoint Paths
This paper revisits the classical Edge Disjoint Paths (EDP) problem, whe...
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Arcdisjoint Strong Spanning Subdigraphs of Semicomplete Compositions
A strong arc decomposition of a digraph D=(V,A) is a decomposition of it...
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Hierarchical Decompositions of dihypergraphs
In this paper we are interested in decomposing a dihypergraph ℋ = (V, ℰ)...
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The AlmostDisjoint 2Path Decomposition Problem
We consider the problem of decomposing a given (di)graph into paths of l...
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The canonical directed tree decomposition and its applications to the directed disjoint paths problem
The canonical treedecomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the canonical tree decomposition theorem for digraphs. More precisely, we construct directed treedecompositions of digraphs that distinguish all their tangles of order k, for any fixed integer k, in polynomial time. As an application of this canonical treedecomposition theorem, we provide the following result for the directed disjoint paths problem: For every fixed k there is a polynomialtime algorithm which, on input G, and source and terminal vertices (s_1, t_1), …, (s_k, t_k), either 1. determines that there is no set of pairwise vertexdisjoint paths connecting each source s_i to its terminal t_i, or 2.finds a halfintegral solution, i.e., outputs paths P_1, …, P_k such that P_i links s_i to t_i, so that every vertex of the graph is contained in at most two paths. Given known hardness results for the directed disjoint paths problem, our result cannot be improved for general digraphs, neither to fixedparameter tractability nor to fully vertexdisjoint directed paths. As far as we are aware, this is the first time to obtain a tractable result for the kdisjoint paths problem for general digraphs. We expect more applications of our canonical treedecomposition for directed results.
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