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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Vydáno v:	Algorithmica Ročník 82; číslo 6; s. 1616 - 1639
Hlavní autoři:	Araújo, Júlio, Campos, Victor A., Maia, Ana Karolinna, Sau, Ignasi, Silva, Ana
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.06.2020 Springer Nature B.V Springer Verlag
Témata:	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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References	Schrijver (CR27) 2001; 82 Metzlar (CR23) 1993; 57 CR16 Kriesell (CR22) 2005; 95 Bang-Jensen, Gutin (CR2) 2008 Garey, Johnson (CR15) 1979 Diestel (CR11) 2012 Alon, Yuster, Zwick (CR1) 1995; 42 Monien (CR24) 1985; 25 Bodlaender, Jansen, Kratsch (CR6) 2014; 28 Kim, Kim, Ma, Park (CR20) 2018; 88 Sedgewick, Wayne (CR28) 2011 Benhocine, Wojda (CR4) 1983; 7 Downey, Fellows (CR12) 2013 Bang-Jensen, Havet, Maia (CR3) 2015; 562 CR5 CR8 Shachnai, Zehavi (CR29) 2016; 82 Impagliazzo, Paturi, Zane (CR18) 2001; 63 Flum, Grohe (CR13) 2006 Havet, Maia, Mohar (CR17) 2018; 87 CR21 Oxley (CR26) 1992 Niedermeier (CR25) 2006 Slivkins (CR30) 2010; 24 Zehavi (CR32) 2016; 116 Brewster, Hell, Pantel, Rizzi, Yeo (CR7) 2003; 44 Cohen, Havet, Lochet, Nisse (CR9) 2018; 89 Venkatesan, Pandu Rangan (CR31) 1996; 60 Cygan, Fomin, Kowalik, Lokshtanov, Marx, Pilipczuk, Pilipczuk, Saurabh (CR10) 2015 Itai, Perl, Shiloach (CR19) 1982; 12 Fomin, Lokshtanov, Panolan, Saurabh (CR14) 2016; 63 RC Brewster (659_CR7) 2003; 44 H Shachnai (659_CR29) 2016; 82 J Bang-Jensen (659_CR3) 2015; 562 J Bang-Jensen (659_CR2) 2008 R Diestel (659_CR11) 2012 RG Downey (659_CR12) 2013 A Metzlar (659_CR23) 1993; 57 659_CR5 659_CR16 M Zehavi (659_CR32) 2016; 116 659_CR8 R Impagliazzo (659_CR18) 2001; 63 A Schrijver (659_CR27) 2001; 82 N Alon (659_CR1) 1995; 42 HL Bodlaender (659_CR6) 2014; 28 R Niedermeier (659_CR25) 2006 FV Fomin (659_CR14) 2016; 63 J Flum (659_CR13) 2006 JG Oxley (659_CR26) 1992 R Kim (659_CR20) 2018; 88 N Cohen (659_CR9) 2018; 89 MR Garey (659_CR15) 1979 G Venkatesan (659_CR31) 1996; 60 R Sedgewick (659_CR28) 2011 A Slivkins (659_CR30) 2010; 24 M Cygan (659_CR10) 2015 B Monien (659_CR24) 1985; 25 A Itai (659_CR19) 1982; 12 F Havet (659_CR17) 2018; 87 A Benhocine (659_CR4) 1983; 7 659_CR21 M Kriesell (659_CR22) 2005; 95
References_xml	– year: 2006 ident: CR13 publication-title: Parameterized Complexity Theory – volume: 28 start-page: 277 issue: 1 year: 2014 end-page: 305 ident: CR6 article-title: Kernelization lower bounds by cross-composition publication-title: SIAM J. Discrete Math. doi: 10.1137/120880240 – volume: 562 start-page: 283 year: 2015 end-page: 303 ident: CR3 article-title: Finding a subdivision of a digraph publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2014.10.004 – year: 2012 ident: CR11 publication-title: Graph Theory, Graduate Texts in Mathematics – year: 2013 ident: CR12 publication-title: Fundamentals of Parameterized Complexity doi: 10.1007/978-1-4471-5559-1 – volume: 88 start-page: 592 issue: 4 year: 2018 end-page: 605 ident: CR20 article-title: Cycles with two blocks in -chromatic digraphs publication-title: J. Graph Theory doi: 10.1002/jgt.22232 – volume: 57 start-page: 228 issue: 2 year: 1993 end-page: 238 ident: CR23 article-title: Disjoint paths in acyclic digraphs publication-title: J. Comb. Theory, Ser. B doi: 10.1006/jctb.1993.1018 – volume: 63 start-page: 512 issue: 4 year: 2001 end-page: 530 ident: CR18 article-title: Which problems have strongly exponential complexity? publication-title: J. Comput. Syst. Sci. doi: 10.1006/jcss.2001.1774 – ident: CR16 – volume: 95 start-page: 168 issue: 1 year: 2005 end-page: 2005 ident: CR22 article-title: Disjoint -paths in digraphs publication-title: J. Comb. Theory, Ser. B doi: 10.1016/j.jctb.2005.03.001 – volume: 44 start-page: 81 issue: 2 year: 2003 end-page: 94 ident: CR7 article-title: Packing paths in digraphs publication-title: J. Graph Theory doi: 10.1002/jgt.10126 – year: 2015 ident: CR10 publication-title: Parameterized Algorithms doi: 10.1007/978-3-319-21275-3 – volume: 82 start-page: 319 issue: 2 year: 2001 end-page: 321 ident: CR27 article-title: A short proof of Mader’s -paths theorem publication-title: J. Comb. Theory, Ser. B doi: 10.1006/jctb.2000.2029 – year: 2011 ident: CR28 publication-title: Algorithms – volume: 42 start-page: 844 issue: 4 year: 1995 end-page: 856 ident: CR1 article-title: Color-coding publication-title: J. ACM doi: 10.1145/210332.210337 – volume: 60 start-page: 249 year: 1996 end-page: 253 ident: CR31 article-title: Approximate triclique coloring for register allocation publication-title: Inf. Process. Lett. doi: 10.1016/S0020-0190(96)00174-3 – volume: 24 start-page: 146 issue: 1 year: 2010 end-page: 157 ident: CR30 article-title: Parameterized tractability of edge-disjoint paths on directed acyclic graphs publication-title: SIAM J. Discrete Math. doi: 10.1137/070697781 – ident: CR8 – volume: 89 start-page: 439 issue: 4 year: 2018 end-page: 456 ident: CR9 article-title: Subdivisions of oriented cycles in digraphs with large chromatic number publication-title: J. Graph Theory doi: 10.1002/jgt.22360 – year: 1992 ident: CR26 publication-title: Matroid Theory – year: 1979 ident: CR15 publication-title: Computers and Intractability. A guide to the Theory of NP-Completeness – year: 2008 ident: CR2 publication-title: Digraphs: Theory, Algorithms and Applications – volume: 7 start-page: 469 issue: 4 year: 1983 end-page: 473 ident: CR4 article-title: On the existence of specified cycles in a tournament publication-title: J. Graph Theory doi: 10.1002/jgt.3190070413 – ident: CR21 – volume: 87 start-page: 536 issue: 4 year: 2018 end-page: 560 ident: CR17 article-title: Finding a subdivision of a prescribed digraph of order 4 publication-title: J. Graph Theory doi: 10.1002/jgt.22174 – volume: 82 start-page: 488 issue: 3 year: 2016 end-page: 502 ident: CR29 article-title: Representative families: a unified tradeoff-based approach publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2015.11.008 – year: 2006 ident: CR25 publication-title: Invitation to Fixed-Parameter Algorithms doi: 10.1093/acprof:oso/9780198566076.001.0001 – volume: 25 start-page: 239 year: 1985 end-page: 254 ident: CR24 article-title: How to find long paths efficiently publication-title: Ann. Discrete Math. – volume: 12 start-page: 277 issue: 3 year: 1982 end-page: 286 ident: CR19 article-title: The complexity of finding maximum disjoint paths with length constraints publication-title: Networks doi: 10.1002/net.3230120306 – ident: CR5 – volume: 63 start-page: 29:1 issue: 4 year: 2016 end-page: 29:60 ident: CR14 article-title: Efficient computation of representative families with applications in parameterized and exact algorithms publication-title: J. ACM doi: 10.1145/2886094 – volume: 116 start-page: 419 issue: 6 year: 2016 end-page: 422 ident: CR32 article-title: A randomized algorithm for long directed cycle publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2016.02.005 – volume-title: Parameterized Complexity Theory year: 2006 ident: 659_CR13 – volume-title: Invitation to Fixed-Parameter Algorithms year: 2006 ident: 659_CR25 doi: 10.1093/acprof:oso/9780198566076.001.0001 – volume: 7 start-page: 469 issue: 4 year: 1983 ident: 659_CR4 publication-title: J. Graph Theory doi: 10.1002/jgt.3190070413 – volume: 95 start-page: 168 issue: 1 year: 2005 ident: 659_CR22 publication-title: J. Comb. Theory, Ser. B doi: 10.1016/j.jctb.2005.03.001 – volume-title: Fundamentals of Parameterized Complexity year: 2013 ident: 659_CR12 doi: 10.1007/978-1-4471-5559-1 – volume-title: Computers and Intractability. A guide to the Theory of NP-Completeness year: 1979 ident: 659_CR15 – volume: 28 start-page: 277 issue: 1 year: 2014 ident: 659_CR6 publication-title: SIAM J. Discrete Math. doi: 10.1137/120880240 – volume-title: Graph Theory, Graduate Texts in Mathematics year: 2012 ident: 659_CR11 – ident: 659_CR21 doi: 10.1007/11917496_6 – volume: 24 start-page: 146 issue: 1 year: 2010 ident: 659_CR30 publication-title: SIAM J. Discrete Math. doi: 10.1137/070697781 – volume: 25 start-page: 239 year: 1985 ident: 659_CR24 publication-title: Ann. Discrete Math. – volume: 42 start-page: 844 issue: 4 year: 1995 ident: 659_CR1 publication-title: J. ACM doi: 10.1145/210332.210337 – volume-title: Matroid Theory year: 1992 ident: 659_CR26 – volume: 57 start-page: 228 issue: 2 year: 1993 ident: 659_CR23 publication-title: J. Comb. Theory, Ser. B doi: 10.1006/jctb.1993.1018 – volume: 82 start-page: 488 issue: 3 year: 2016 ident: 659_CR29 publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2015.11.008 – volume: 89 start-page: 439 issue: 4 year: 2018 ident: 659_CR9 publication-title: J. Graph Theory doi: 10.1002/jgt.22360 – volume: 88 start-page: 592 issue: 4 year: 2018 ident: 659_CR20 publication-title: J. Graph Theory doi: 10.1002/jgt.22232 – volume: 87 start-page: 536 issue: 4 year: 2018 ident: 659_CR17 publication-title: J. Graph Theory doi: 10.1002/jgt.22174 – volume: 82 start-page: 319 issue: 2 year: 2001 ident: 659_CR27 publication-title: J. Comb. Theory, Ser. B doi: 10.1006/jctb.2000.2029 – ident: 659_CR5 doi: 10.1007/978-3-540-27836-8_21 – ident: 659_CR16 doi: 10.1145/1993636.1993700 – volume: 60 start-page: 249 year: 1996 ident: 659_CR31 publication-title: Inf. Process. Lett. doi: 10.1016/S0020-0190(96)00174-3 – volume: 63 start-page: 512 issue: 4 year: 2001 ident: 659_CR18 publication-title: J. Comput. Syst. Sci. doi: 10.1006/jcss.2001.1774 – volume: 63 start-page: 29:1 issue: 4 year: 2016 ident: 659_CR14 publication-title: J. ACM doi: 10.1145/2886094 – volume-title: Algorithms year: 2011 ident: 659_CR28 – volume: 44 start-page: 81 issue: 2 year: 2003 ident: 659_CR7 publication-title: J. Graph Theory doi: 10.1002/jgt.10126 – ident: 659_CR8 doi: 10.1007/978-3-662-53536-3_6 – volume-title: Digraphs: Theory, Algorithms and Applications year: 2008 ident: 659_CR2 – volume: 116 start-page: 419 issue: 6 year: 2016 ident: 659_CR32 publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2016.02.005 – volume: 562 start-page: 283 year: 2015 ident: 659_CR3 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2014.10.004 – volume-title: Parameterized Algorithms year: 2015 ident: 659_CR10 doi: 10.1007/978-3-319-21275-3 – volume: 12 start-page: 277 issue: 3 year: 1982 ident: 659_CR19 publication-title: Networks doi: 10.1002/net.3230120306
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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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