2017/10/25 by Michael Etscheid, Matthias Mnich · 12 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Graph Labeling and Dimension Problems #Parameterized complexity #Time complexity #Theory of computation #Mathematics #Combinatorics #Algorithm #Vertex cover #Graph #Upper and lower bounds #Running time #Discrete mathematics
paper · pdf · doi:10.1007/s00453-017-0388-z
published in Algorithmica 80(9), 2574-2615 (Springer Science+Business Media)
openalex publication_date 2017/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 \(8k⋅ O(m)\) ; this significantly improves the previous analysis with run time \(8k⋅ O(n4)\) . We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards−Erd ő s 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 ).