Linear Kernels and Linear-Time Algorithms for Finding Large Cuts

Michael Etscheid, Matthias Mnich

Research output: Contribution to journalArticleAcademicpeer-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. (Algorithmica 72(3):734-757, 2015) for (Signed) Max-Cut Above Edwards-Erd A s Bound can be implemented so as to run in linear time ; this significantly improves the previous analysis with run time .

We give an asymptotically optimal kernel for (Signed) Max-Cut Above Edwards-Erd A s Bound with O(k) vertices, improving a kernel with vertices by Crowston et al. (Theor Comput Sci 513:53-64, 2013).

We improve all known kernels for parameterizations above strongly -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 vertices to O(k) vertices.

Therefore, Max Acyclic Subdigraph parameterized above Poljak-Turzik bound admits a kernel with O(k) vertices and can be solved in time; this answers an open question by Crowston et al. (Proceedings of FSTTCS 2012, Leibniz international proceedings in informatics, Hyderabad, 2012).

Original languageEnglish
Pages (from-to)2574-2615
Number of pages42
JournalAlgorithmica
Volume80
Issue number9
Early online date2017
DOIs
Publication statusPublished - Sep 2018

Keywords

  • Max-cut
  • Kernelization
  • Linear-time algorithms
  • MAX-CUT
  • HALF
  • COMPLEXITY

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