Linear Kernels and Linear-Time Algorithms for Finding Large Cuts

Michael Etscheid, Matthias Mnich

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingAcademicpeer-review

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. [ICALP 2012] for (SIGNED) MAX-CUT ABOVE EDWARDS-ERDO{combining double acute accent}S BOUND can be implemented in such a way that it runs 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-ERDO{combining double acute accent}S BOUND with O(k) vertices, improving a kernel with O(k3) vertices by Crowston et al. [COCOON 2013]. ▪ We improve all known kernels for strongly λ-extendable properties parameterized above tight lower bound by Crowston et al. [FSTTCS 2013] from O(k3) vertices to O(k) vertices. ▪ As a consequence, MAX ACYCLIC SUBDIGRAPH parameterized above Poljak-Turzík bound admits a kernel with O(k) vertices and can be solved in time 2O(k)·nO(1); this answers an open question by Crowston et al. [FSTTCS 2012]. All presented kernels can be computed in time O(km).

Original languageEnglish
Title of host publication27th International Symposium on Algorithms and Computation (ISAAC 2016)
PublisherSchloss Dagstuhl
DOIs
Publication statusPublished - 2016

Publication series

SeriesLeibniz International Proceedings in Informatics (LIPIcs)
Volume64

Keywords

  • Max-Cut
  • fixed-parameter tractability
  • kernelization

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