A Polynomial Kernel for Funnel Arc Deletion Set

In Directed Feedback Arc Set ( DFAS ) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C - Arc Deletion Set ( C - ADS ), a variant of D...

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Published in:	Algorithmica Vol. 84; no. 8; pp. 2358 - 2378
Main Author:	Garlet Milani, Marcelo
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2022 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Deletion Funnels Graph theory Kernels Mathematics of Computing Polynomials Special Issue: Parameterized and Exact Computation (IPEC 2020) Theory of Computation Graph editing Kernels Funnels Parameterized algorithm Directed feedback arc set
ISSN:	0178-4617, 1432-0541
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Abstract	In Directed Feedback Arc Set ( DFAS ) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C - Arc Deletion Set ( C - ADS ), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class C . In this work, we choose C to be the class of funnels . Funnel - ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k . So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel - ADS with O ( k 6 ) many vertices and O ( k 7 ) many arcs, computable in O ( n m ) time, where n is the number of vertices and m the number of arcs in the input digraph.
AbstractList	In Directed Feedback Arc Set ( DFAS ) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider $$\mathcal {C}$$ C - Arc Deletion Set ( $$\mathcal {C}$$ C - ADS ), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class $$\mathcal {C}$$ C . In this work, we choose $$\mathcal {C}$$ C to be the class of funnels . Funnel - ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k . So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel - ADS with $$\mathcal {O}(k^6)$$ O ( k 6 ) many vertices and $$\mathcal {O}(k^7)$$ O ( k 7 ) many arcs, computable in $$\mathcal {O}(nm)$$ O ( n m ) time, where n is the number of vertices and m the number of arcs in the input digraph. In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C-Arc Deletion Set (C-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class C. In this work, we choose C to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel-ADS with O(k6) many vertices and O(k7) many arcs, computable in O(nm) time, where n is the number of vertices and m the number of arcs in the input digraph. In Directed Feedback Arc Set ( DFAS ) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider C - Arc Deletion Set ( C - ADS ), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class C . In this work, we choose C to be the class of funnels . Funnel - ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k . So far no polynomial kernels for this problem were known. Our main result is a kernel for Funnel - ADS with O ( k 6 ) many vertices and O ( k 7 ) many arcs, computable in O ( n m ) time, where n is the number of vertices and m the number of arcs in the input digraph.
Author	Garlet Milani, Marcelo
Author_xml	– sequence: 1 givenname: Marcelo orcidid: 0000-0001-8398-4751 surname: Garlet Milani fullname: Garlet Milani, Marcelo email: m.garletmillani@tu-berlin.de organization: Logic and Semantics, Institute of Software Engineering and Theoretical Computer Science
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References	ChenJLiuYLuSO’sullivanBRazgonIA fixed-parameter algorithm for the directed feedback vertex set problemJ. ACM200855521245654610.1145/1411509.1411511 Abu-KhzamFNA kernelization algorithm for d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-hitting setJ. Comput. Syst. Sci.2010767524531266281710.1016/j.jcss.2009.09.002 Garlet MillaniMMolterHNiedermeierRSorgeMEfficient algorithms for measuring the funnel-likeness of dagsJ. Combin. Optim.2020391216245404710510.1007/s10878-019-00464-4 CyganMFominFVKowalikŁLokshtanovDMarxDPilipczukMPilipczukMSaurabhSParameterized Algorithms2015ChamSpringer10.1007/978-3-319-21275-3 DowneyRGFellowsMRFundamentals of Parameterized Complexity2013LondonSpringer10.1007/978-1-4471-5559-1 FominFVLokshtanovDSaurabhSZehaviMKernelization: Theory of Parameterized Preprocessing2019United KingdomCambridge University Press1426.68003 Bang-JensenJMaddaloniASaurabhSAlgorithms and kernels for feedback set problems in generalizations of tournamentsAlgorithmica2016762320343353702110.1007/s00453-015-0038-2 Lokshtanov, D., Ramanujan, M., Saurabh, S., Sharma, R., Zehavi, M.: Wannabe bounded treewidth graphs admit a polynomial kernel for dfvs. In: Workshop on Algorithms and Data Structures, pp. 523–537 (2019). Springer Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization—preprocessing with a guarantee. In: The Multivariate Algorithmic Revolution and Beyond, pp. 129–161. Springer, Berlin, Heidelberg (2012) AgrawalASaurabhSSharmaRZehaviMKernels for deletion to classes of acyclic digraphsJ. Comput. Syst. Sci.201892921372139710.1016/j.jcss.2017.07.008 MnichMvan LeeuwenEJPolynomial kernels for deletion to classes of acyclic digraphsDiscrete Optim.2017254876367987410.1016/j.disopt.2017.02.002 Kratsch, S.: Recent developments in kernelization: a survey. Bull. EATCS 2(113) (2014) 960_CR6 A Agrawal (960_CR8) 2018; 92 FV Fomin (960_CR12) 2019 J Chen (960_CR3) 2008; 55 M Mnich (960_CR7) 2017; 25 RG Downey (960_CR2) 2013 FN Abu-Khzam (960_CR4) 2010; 76 M Garlet Millani (960_CR9) 2020; 39 M Cygan (960_CR1) 2015 J Bang-Jensen (960_CR5) 2016; 76 960_CR10 960_CR11
References_xml	– reference: Kratsch, S.: Recent developments in kernelization: a survey. Bull. EATCS 2(113) (2014) – reference: FominFVLokshtanovDSaurabhSZehaviMKernelization: Theory of Parameterized Preprocessing2019United KingdomCambridge University Press1426.68003 – reference: Lokshtanov, D., Ramanujan, M., Saurabh, S., Sharma, R., Zehavi, M.: Wannabe bounded treewidth graphs admit a polynomial kernel for dfvs. In: Workshop on Algorithms and Data Structures, pp. 523–537 (2019). Springer – reference: MnichMvan LeeuwenEJPolynomial kernels for deletion to classes of acyclic digraphsDiscrete Optim.2017254876367987410.1016/j.disopt.2017.02.002 – reference: Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization—preprocessing with a guarantee. In: The Multivariate Algorithmic Revolution and Beyond, pp. 129–161. Springer, Berlin, Heidelberg (2012) – reference: DowneyRGFellowsMRFundamentals of Parameterized Complexity2013LondonSpringer10.1007/978-1-4471-5559-1 – reference: AgrawalASaurabhSSharmaRZehaviMKernels for deletion to classes of acyclic digraphsJ. Comput. Syst. Sci.201892921372139710.1016/j.jcss.2017.07.008 – reference: CyganMFominFVKowalikŁLokshtanovDMarxDPilipczukMPilipczukMSaurabhSParameterized Algorithms2015ChamSpringer10.1007/978-3-319-21275-3 – reference: ChenJLiuYLuSO’sullivanBRazgonIA fixed-parameter algorithm for the directed feedback vertex set problemJ. ACM200855521245654610.1145/1411509.1411511 – reference: Abu-KhzamFNA kernelization algorithm for d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-hitting setJ. Comput. Syst. Sci.2010767524531266281710.1016/j.jcss.2009.09.002 – reference: Garlet MillaniMMolterHNiedermeierRSorgeMEfficient algorithms for measuring the funnel-likeness of dagsJ. Combin. Optim.2020391216245404710510.1007/s10878-019-00464-4 – reference: Bang-JensenJMaddaloniASaurabhSAlgorithms and kernels for feedback set problems in generalizations of tournamentsAlgorithmica2016762320343353702110.1007/s00453-015-0038-2 – volume: 25 start-page: 48 year: 2017 ident: 960_CR7 publication-title: Discrete Optim. doi: 10.1016/j.disopt.2017.02.002 – volume: 92 start-page: 9 year: 2018 ident: 960_CR8 publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2017.07.008 – ident: 960_CR10 – volume-title: Fundamentals of Parameterized Complexity year: 2013 ident: 960_CR2 doi: 10.1007/978-1-4471-5559-1 – volume: 55 start-page: 21 issue: 5 year: 2008 ident: 960_CR3 publication-title: J. ACM doi: 10.1145/1411509.1411511 – volume: 39 start-page: 216 issue: 1 year: 2020 ident: 960_CR9 publication-title: J. Combin. Optim. doi: 10.1007/s10878-019-00464-4 – ident: 960_CR6 doi: 10.1007/978-3-030-24766-9_38 – volume: 76 start-page: 320 issue: 2 year: 2016 ident: 960_CR5 publication-title: Algorithmica doi: 10.1007/s00453-015-0038-2 – volume: 76 start-page: 524 issue: 7 year: 2010 ident: 960_CR4 publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2009.09.002 – ident: 960_CR11 doi: 10.1007/978-3-642-30891-8_10 – volume-title: Parameterized Algorithms year: 2015 ident: 960_CR1 doi: 10.1007/978-3-319-21275-3 – volume-title: Kernelization: Theory of Parameterized Preprocessing year: 2019 ident: 960_CR12
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Snippet	In Directed Feedback Arc Set ( DFAS ) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem... In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Deletion Funnels Graph theory Kernels Mathematics of Computing Polynomials Special Issue: Parameterized and Exact Computation (IPEC 2020) Theory of Computation
Title	A Polynomial Kernel for Funnel Arc Deletion Set
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