On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

For two positive integers k and ℓ , a ( k × ℓ ) - spindle is the union of k pairwise internally vertex-disjoint directed paths with ℓ arcs each between two vertices u and v . We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contai...

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Bibliographic Details
Published in:Algorithmica Vol. 82; no. 6; pp. 1616 - 1639
Main Authors: Araújo, Júlio, Campos, Victor A., Maia, Ana Karolinna, Sau, Ignasi, Silva, Ana
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2020
Springer Nature B.V
Springer Verlag
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Color coding
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Mathematics
Mathematics of Computing
Polynomials
Spindles
Theory of Computation
G.2.2
Complexity dichotomy
Spindle
Representative family
FPT algorithm
Digraph subdivision
Parameterized complexity
F.2.2
parameterized complexity
complexity dichotomy
spindle
representative family
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:For two positive integers k and ℓ , a ( k × ℓ ) - spindle is the union of k pairwise internally vertex-disjoint directed paths with ℓ arcs each between two vertices u and v . We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed ℓ ≥ 1 , finding the largest k such that an input digraph G contains a subdivision of a ( k × ℓ ) -spindle is polynomial-time solvable if ℓ ≤ 3 , and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00659-5