Linear Kernels and Linear-Time Algorithms for Finding Large Cuts

The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph....

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Vydané v:	Algorithmica Ročník 80; číslo 9; s. 2574 - 2615
Hlavní autori:	Etscheid, Michael, Mnich, Matthias
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.09.2018 Springer Nature B.V
Predmet:	Algorithm Analysis and Problem Complexity Algorithms Combinatorial analysis Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Informatics Kernels Lower bounds Mathematics of Computing Run time (computers) Special Issue dedicated to the 60th Birthday of Gregory Gutin Theory of Computation 68W05 68R10 Linear-time algorithms 05C85 Max-cut Kernelization
ISSN:	0178-4617, 1432-0541
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Abstract	The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results: We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757, 2015 ) for (Signed) Max-Cut Above Edwards−Erd ő s Bound can be implemented so as to run in linear time 8 k · O ( m ) ; this significantly improves the previous analysis with run time 8 k · O ( n 4 ) . We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erd ő s Bound with O ( k ) vertices, improving a kernel with O ( k 3 ) vertices by Crowston et al. (Theor Comput Sci 513:53–64, 2013 ). We improve all known kernels for parameterizations above strongly λ -extendible properties (a generalization of the Max-Cut results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati, 2013 ) from O ( k 3 ) vertices to O ( k ) vertices. Therefore, Max Acyclic Subdigraph parameterized above Poljak–Turzík bound admits a kernel with O ( k ) vertices and can be solved in 2 O ( k ) · n O ( 1 ) time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012 ). All presented kernels can be computed in time O ( km ).
AbstractList	The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results:We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757, 2015) for (Signed) Max-Cut Above Edwards−Erdős Bound can be implemented so as to run in linear time8k·O(m); this significantly improves the previous analysis with run time 8k·O(n4).We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erdős Bound with O(k) vertices, improving a kernel with O(k3) vertices by Crowston et al. (Theor Comput Sci 513:53–64, 2013).We improve all known kernels for parameterizations above strongly λ-extendible properties (a generalization of the Max-Cut results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati, 2013) from O(k3) vertices to O(k) vertices.Therefore, Max Acyclic Subdigraph parameterized above Poljak–Turzík bound admits a kernel with O(k) vertices and can be solved in 2O(k)·nO(1) time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012). All presented kernels can be computed in time O(km). The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of those results: We show that an algorithm by Crowston et al. (Algorithmica 72(3):734–757, 2015 ) for (Signed) Max-Cut Above Edwards−Erd ő s Bound can be implemented so as to run in linear time 8 k · O ( m ) ; this significantly improves the previous analysis with run time 8 k · O ( n 4 ) . We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erd ő s Bound with O ( k ) vertices, improving a kernel with O ( k 3 ) vertices by Crowston et al. (Theor Comput Sci 513:53–64, 2013 ). We improve all known kernels for parameterizations above strongly λ -extendible properties (a generalization of the Max-Cut results) by Crowston et al. (Proceedings of FSTTCS 2013, Leibniz international proceedings in informatics, Guwahati, 2013 ) from O ( k 3 ) vertices to O ( k ) vertices. Therefore, Max Acyclic Subdigraph parameterized above Poljak–Turzík bound admits a kernel with O ( k ) vertices and can be solved in 2 O ( k ) · n O ( 1 ) time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012 ). All presented kernels can be computed in time O ( km ).
Author	Etscheid, Michael Mnich, Matthias
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Cites_doi	10.1016/j.ipl.2010.08.001 10.1016/j.tcs.2005.10.010 10.1016/j.jcss.2008.08.004 10.1016/j.ipl.2007.05.014 10.1016/0012-365X(86)90192-5 10.1007/s10878-009-9212-2 10.1088/0305-4470/15/10/028 10.1137/16M1061862 10.1016/j.jcss.2014.04.011 10.4153/CJM-1973-048-x 10.1016/j.jcss.2011.01.004 10.1006/jcss.2001.1774 10.1007/s00224-007-1330-6 10.1007/s00453-014-9870-z 10.1017/S0963548300000596 10.1109/TCAD.2006.882642 10.1016/j.tcs.2013.10.026 10.1016/S0304-3975(99)00150-4 10.1093/imaman/13.3.201 10.1002/bs.3830040405 10.1016/j.jcss.2013.10.002 10.1007/BF01456961 10.1137/140962838 10.1007/s00453-010-9428-7 10.1090/dimacs/020/04 10.1007/978-1-4684-2001-2_9 10.1007/978-3-662-47672-7_76
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Snippet	The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Combinatorial analysis Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Informatics Kernels Lower bounds Mathematics of Computing Run time (computers) Special Issue dedicated to the 60th Birthday of Gregory Gutin Theory of Computation
Title	Linear Kernels and Linear-Time Algorithms for Finding Large Cuts
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