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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Vydané v:	Algorithmica Ročník 82; číslo 6; s. 1616 - 1639
Hlavní autori:	Araújo, Júlio, Campos, Victor A., Maia, Ana Karolinna, Sau, Ignasi, Silva, Ana
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2020 Springer Nature B.V Springer Verlag
Predmet:	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
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Abstract	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.
AbstractList	For two positive integers k and , a (k ×)-spindle is the union of k pairwise internally vertex-disjoint directed paths with 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 ≥ 1, finding the largest k such that an input digraph G contains a subdivision of a (k ×)-spindle is polynomial-time solvable if ≤ 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. 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. 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.
Author	Campos, Victor A. Araújo, Júlio Maia, Ana Karolinna Sau, Ignasi Silva, Ana
Author_xml	– sequence: 1 givenname: Júlio surname: Araújo fullname: Araújo, Júlio organization: Departamento de Matemática, ParGO Research Group, Universidade Federal do Ceará – sequence: 2 givenname: Victor A. surname: Campos fullname: Campos, Victor A. organization: Departamento de Computação, ParGO Research Group, Universidade Federal do Ceará – sequence: 3 givenname: Ana Karolinna surname: Maia fullname: Maia, Ana Karolinna organization: Departamento de Computação, ParGO Research Group, Universidade Federal do Ceará – sequence: 4 givenname: Ignasi orcidid: 0000-0002-8981-9287 surname: Sau fullname: Sau, Ignasi email: ignasi.sau@lirmm.fr organization: Departamento de Matemática, ParGO Research Group, Universidade Federal do Ceará, CNRS, AlGCo Project Team, LIRMM, Université de Montpellier – sequence: 5 givenname: Ana surname: Silva fullname: Silva, Ana organization: Departamento de Matemática, ParGO Research Group, Universidade Federal do Ceará
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Cites_doi	10.1137/120880240 10.1016/j.tcs.2014.10.004 10.1007/978-1-4471-5559-1 10.1002/jgt.22232 10.1006/jctb.1993.1018 10.1006/jcss.2001.1774 10.1016/j.jctb.2005.03.001 10.1002/jgt.10126 10.1007/978-3-319-21275-3 10.1006/jctb.2000.2029 10.1145/210332.210337 10.1016/S0020-0190(96)00174-3 10.1137/070697781 10.1002/jgt.22360 10.1002/jgt.3190070413 10.1002/jgt.22174 10.1016/j.jcss.2015.11.008 10.1093/acprof:oso/9780198566076.001.0001 10.1002/net.3230120306 10.1145/2886094 10.1016/j.ipl.2016.02.005 10.1007/11917496_6 10.1007/978-3-540-27836-8_21 10.1145/1993636.1993700 10.1007/978-3-662-53536-3_6
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Keywords	G.2.2 Complexity dichotomy Spindle Representative family FPT algorithm Digraph subdivision Parameterized complexity F.2.2 parameterized complexity complexity dichotomy spindle representative family
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Snippet	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... 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... For two positive integers k and , a (k ×)-spindle is the union of k pairwise internally vertex-disjoint directed paths with arcs each between two vertices u...
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SubjectTerms	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
Title	On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths
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