Linear Kernels and Linear-Time Algorithms for Finding Large Cuts

• Published in: Algorithmica 2018-09, Vol.80 (9), p.2574-2615
• Main Authors: Etscheid, Michael, Mnich, Matthias
• Format: Article
• Language: English
• Subjects: 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
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-017-0388-z

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 ).