A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection problem asks if the vertices of D can be partitioned into two parts, say L and R , such that | L | and | R | differ by at most 1 and the number of arcs from R to L is at most k . This problem is known to be...

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Vydané v:	Algorithmica Ročník 83; číslo 6; s. 1861 - 1884
Hlavní autori:	Madathil, Jayakrishnan, Sharma, Roohani, Zehavi, Meirav
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2021 Springer Nature B.V
Predmet:	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Kernels Mathematics of Computing Polynomials Theory of Computation Bisection Polynomial kernel Splitters FPT Algorithm Chromatic coding Semicomplete digraph Tournament
ISSN:	0178-4617, 1432-0541
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Abstract	Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection problem asks if the vertices of D can be partitioned into two parts, say L and R , such that \| L \| and \| R \| differ by at most 1 and the number of arcs from R to L is at most k . This problem is known to be NP-hard even when k = 0 . We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that M inimum D irected B isection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that M inimum D irected B isection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use ( n , k , k 2 ) -splitters, which, to the best of our knowledge, have never been used before in the design of kernels. We also prove that M inimum D irected B isection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.
AbstractList	Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that \|L\| and \|R\| differ by at most 1 and the number of arcs from R to L is at most k. This problem is known to be NP-hard even when k=0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n,k,k2)-splitters, which, to the best of our knowledge, have never been used before in the design of kernels. We also prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments. Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection problem asks if the vertices of D can be partitioned into two parts, say L and R , such that \| L \| and \| R \| differ by at most 1 and the number of arcs from R to L is at most k . This problem is known to be NP-hard even when k = 0 . We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that M inimum D irected B isection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that M inimum D irected B isection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use ( n , k , k 2 ) -splitters, which, to the best of our knowledge, have never been used before in the design of kernels. We also prove that M inimum D irected B isection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.
Author	Sharma, Roohani Madathil, Jayakrishnan Zehavi, Meirav
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Cites_doi	10.1007/978-3-662-44465-8_22 10.1137/1.9781611975482.64 10.1016/j.jcss.2010.10.001 10.1137/0221016 10.1016/0304-3975(76)90059-1 10.1016/j.jctb.2019.01.006 10.1016/j.disopt.2010.03.010 10.1137/140988553 10.1016/j.jda.2009.08.001 10.1002/jgt.22408 10.1145/210332.210337 10.1007/978-3-662-53622-3_7 10.1145/1374376.1374415 10.1007/978-3-642-17517-6_3 10.1007/BF01305310 10.1007/978-3-642-02927-1_6 10.1007/978-3-642-45043-3_8 10.1137/S009753970139567X 10.1016/j.jctb.2014.12.005 10.1007/s00453-014-9928-y 10.1016/j.tcs.2005.10.007 10.1145/3196276 10.1016/j.jctb.2010.10.003 10.1007/978-3-319-21275-3 10.1016/S0020-0190(03)00251-5
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In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23–25 October 1995, pp. 182–191 (1995) CyganMLokshtanovDPilipczukMPilipczukMSaurabhSMinimum bisection is fixed-parameter tractableSIAM J. Comput.2019482417450393792510.1137/140988553 Feige, U.: Faster fast (feedback arc set in tournaments). CoRR arXiv:abs/0911.5094 (2009) Karpinski, M., Schudy, W.: Faster algorithms for feedback arc set tournament, kemeny rank aggregation and betweenness tournament. In: Algorithms and Computation—21st International Symposium, ISAAC 2010, Jeju Island, Korea, December 15–17, 2010, Proceedings, Part I, pp. 3–14 (2010) LampisMKaouriGMitsouVOn the algorithmic effectiveness of digraph decompositions and complexity measuresDiscrete Optim.201181129138277256610.1016/j.disopt.2010.03.010 AlonNYusterRZwickUColor-codingJ. ACM1995424844856141178710.1145/210332.210337 BessySFominFVGaspersSPaulCPerezASaurabhSThomasséSKernels for feedback arc set in tournamentsJ. Comput. Syst. Sci.201177610711078285801010.1016/j.jcss.2010.10.001 KimISeymourPDTournament minorsJ. Comb. Theory Ser. B2015112138153332303910.1016/j.jctb.2014.12.005 Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5–12, 2009, Proceedings, Part I, pp. 49–58 (2009) RédeiLEin kombinatorischer satzSatz. Acta Litt. Szeged1934739430009.14606 CamionPChemins et circuits hamiltoniens des graphes completsComptes Rendus Hebdomadaires des Séances de l’Académie des Sciences195924921215121521227350092.15801 BarberoFPaulCPilipczukMExploring the complexity of layout parameters in tournaments and semicomplete digraphsACM Trans. Algorithms201814338:138:31384134010.1145/3196276 PapadimitriouCHSideriMThe bisection width of grid graphsMath. Syst. Theory199629297110136879310.1007/BF01305310 Bang-JensenJGutinGZDigraphs—Theory, Algorithms and Applications2002BerlinSpringer1001.05002 DomMGuoJHüffnerFNiedermeierRTrußAFixed-parameter tractability results for feedback set problems in tournamentsJ. Discrete Algorithms2010817686255888110.1016/j.jda.2009.08.001 GareyMRJohnsonDSStockmeyerLJSome simplified NP-complete graph problemsTheor. Comput. Sci.19761323726741124010.1016/0304-3975(76)90059-1 Diestel, R.: Graph Theory, 4th edn, volume 173 of Graduate texts in mathematics. Springer (2012) FeldmannAEWidmayerPAn o(n4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$^{\text{4}}$$\end{document}) time algorithm to compute the bisection width of solid grid graphsAlgorithmica2015711181200330215110.1007/s00453-014-9928-y CyganMFominFVKowalikLLokshtanovDMarxDPilipczukMPilipczukMSaurabhSParameterized Algorithms2015BerlinSpringer10.1007/978-3-319-21275-3 JansenKKarpinskiMLingasASeidelEPolynomial time approximation schemes for MAX-BISECTION on planar and geometric graphsSIAM J. Comput.2005351110119217880010.1137/S009753970139567X FeigeUYahalomOOn the complexity of finding balanced oneway cutsInf. Process. Lett.200387115197985210.1016/S0020-0190(03)00251-5 Fradkin, A.: Forbidden structures and algorithms in graphs and digraphs. PhD thesis, Princeton University (2011) MarxDParameterized graph separation problemsTheor. Comput. Sci.20063513394406220249810.1016/j.tcs.2005.10.007 BarberoFPaulCPilipczukMStrong immersion is a well-quasi-ordering for semicomplete digraphsJ. Graph Theory2019904484496391518410.1002/jgt.22408 Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17–20, 2008, pp. 255–264 (2008) FominFVPilipczukMOn width measures and topological problems on semi-complete digraphsJ. Comb. Theory Ser. B201913878165397922710.1016/j.jctb.2019.01.006 ChudnovskyMSeymourPDA well-quasi-order for tournamentsJ. Comb. Theory Ser. B201110114753273717710.1016/j.jctb.2010.10.003 Díaz, J., Mertzios, G.B.: Minimum bisection is NP-hard on unit disk graphs. In: Mathematical Foundations of Computer Science 2014—39th International Symposium, MFCS 2014, Budapest, Hungary, August 25–29, 2014. Proceedings, Part II, pp. 251–262 (2014) Madathil, J., Sharma, R., Zehavi, M.: A sub-exponential FPT algorithm and a polynomial kernel for minimum directed bisection on semicomplete digraphs. In: Rossmanith, P., Heggernes, P., Katoen, J.-P. (es) 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26–30, 2019, Aachen, Germany, volume 138 of LIPIcs, pp. 28:1–28:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019) Macgregor, R.M.: On partitioning a graph: a theoretical and empirical study. PhD thesis, University of California, Berkeley (1979) 806_CR30 806_CR32 L Rédei (806_CR33) 1934; 7 806_CR1 K Jansen (806_CR21) 2005; 35 MR Garey (806_CR20) 1976; 1 S Bessy (806_CR6) 2011; 77 TN Bui (806_CR7) 1992; 21 M Chudnovsky (806_CR9) 2011; 101 806_CR19 M Dom (806_CR14) 2010; 8 U Feige (806_CR16) 2003; 87 806_CR15 F Barbero (806_CR5) 2019; 90 M Cygan (806_CR11) 2019; 48 806_CR12 806_CR34 M Lampis (806_CR25) 2011; 8 806_CR13 J Bang-Jensen (806_CR3) 2002 FV Fomin (806_CR18) 2019; 138 P Camion (806_CR8) 1959; 249 CH Papadimitriou (806_CR31) 1996; 29 N Alon (806_CR2) 1995; 42 F Barbero (806_CR4) 2018; 14 I Kim (806_CR24) 2015; 112 806_CR27 806_CR26 806_CR29 AE Feldmann (806_CR17) 2015; 71 806_CR23 806_CR22 M Cygan (806_CR10) 2015 D Marx (806_CR28) 2006; 351
References_xml	– reference: Díaz, J., Mertzios, G.B.: Minimum bisection is NP-hard on unit disk graphs. In: Mathematical Foundations of Computer Science 2014—39th International Symposium, MFCS 2014, Budapest, Hungary, August 25–29, 2014. Proceedings, Part II, pp. 251–262 (2014) – reference: Bang-JensenJGutinGZDigraphs—Theory, Algorithms and Applications2002BerlinSpringer1001.05002 – reference: CamionPChemins et circuits hamiltoniens des graphes completsComptes Rendus Hebdomadaires des Séances de l’Académie des Sciences195924921215121521227350092.15801 – reference: BarberoFPaulCPilipczukMStrong immersion is a well-quasi-ordering for semicomplete digraphsJ. Graph Theory2019904484496391518410.1002/jgt.22408 – reference: AlonNYusterRZwickUColor-codingJ. ACM1995424844856141178710.1145/210332.210337 – reference: PapadimitriouCHSideriMThe bisection width of grid graphsMath. Syst. Theory199629297110136879310.1007/BF01305310 – reference: GareyMRJohnsonDSStockmeyerLJSome simplified NP-complete graph problemsTheor. Comput. Sci.19761323726741124010.1016/0304-3975(76)90059-1 – reference: BarberoFPaulCPilipczukMExploring the complexity of layout parameters in tournaments and semicomplete digraphsACM Trans. Algorithms201814338:138:31384134010.1145/3196276 – reference: Fradkin, A.: Forbidden structures and algorithms in graphs and digraphs. PhD thesis, Princeton University (2011) – reference: Diestel, R.: Graph Theory, 4th edn, volume 173 of Graduate texts in mathematics. Springer (2012) – reference: Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23–25 October 1995, pp. 182–191 (1995) – reference: CyganMFominFVKowalikLLokshtanovDMarxDPilipczukMPilipczukMSaurabhSParameterized Algorithms2015BerlinSpringer10.1007/978-3-319-21275-3 – reference: FeldmannAEWidmayerPAn o(n4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$^{\text{4}}$$\end{document}) time algorithm to compute the bisection width of solid grid graphsAlgorithmica2015711181200330215110.1007/s00453-014-9928-y – reference: BuiTNPeckAPartitioning planar graphsSIAM J. Comput.1992212203215115452010.1137/0221016 – reference: Madathil, J., Sharma, R., Zehavi, M.: A sub-exponential FPT algorithm and a polynomial kernel for minimum directed bisection on semicomplete digraphs. In: Rossmanith, P., Heggernes, P., Katoen, J.-P. (es) 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26–30, 2019, Aachen, Germany, volume 138 of LIPIcs, pp. 28:1–28:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019) – reference: CyganMLokshtanovDPilipczukMPilipczukMSaurabhSMinimum bisection is fixed-parameter tractableSIAM J. Comput.2019482417450393792510.1137/140988553 – reference: Macgregor, R.M.: On partitioning a graph: a theoretical and empirical study. PhD thesis, University of California, Berkeley (1979) – reference: LampisMKaouriGMitsouVOn the algorithmic effectiveness of digraph decompositions and complexity measuresDiscrete Optim.201181129138277256610.1016/j.disopt.2010.03.010 – reference: RédeiLEin kombinatorischer satzSatz. Acta Litt. Szeged1934739430009.14606 – reference: FeigeUYahalomOOn the complexity of finding balanced oneway cutsInf. Process. Lett.200387115197985210.1016/S0020-0190(03)00251-5 – reference: BessySFominFVGaspersSPaulCPerezASaurabhSThomasséSKernels for feedback arc set in tournamentsJ. Comput. Syst. Sci.201177610711078285801010.1016/j.jcss.2010.10.001 – reference: ChudnovskyMSeymourPDA well-quasi-order for tournamentsJ. Comb. Theory Ser. B201110114753273717710.1016/j.jctb.2010.10.003 – reference: Kim, I.: On containment relations in directed graphs. PhD thesis, Princeton University (2013) – reference: Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5–12, 2009, Proceedings, Part I, pp. 49–58 (2009) – reference: van Bevern, R., Feldmann, A.E., Sorge, M., Suchý, O.: On the parameterized complexity of computing graph bisections. In Brandstädt, A., Jansen, K., Reischuk, R. 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Snippet	Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection problem asks if the vertices of D can be partitioned into two... Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts,...
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Title	A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs
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