A sub-exponential FPT algorithm and a polynomial kernel for minimum directed bisection on semicomplete digraphs

Jayakrishnan Madathil, Roohani Sharma, Meirav Zehavi

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

2 Scopus citations

Abstract

Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that |L| and |R| differ by at most 1 and the number of arcs from R to L is at most k. This problem, in general, is W-hard as it is known to be NP-hard even when k = 0. We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that Minimum Directed Bisection on semicomplete digraphs is one of a handful of problems that admit sub-exponential time fixed-parameter tractable algorithms. That is, we show that the problem admits a 2O(k log k)nO(1) time algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use (n, k, k2)-splitters. To the best of our knowledge, this is the first time such pseudorandom objects have been used in the design of kernels. We believe that the framework of designing kernels using splitters could be applied to more problems that admit sub-exponential time algorithms via chromatic coding. To complement the above mentioned results, we prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.

Original languageEnglish
Title of host publication44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019
EditorsJoost-Pieter Katoen, Pinar Heggernes, Peter Rossmanith
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771177
DOIs
StatePublished - 1 Aug 2019
Event44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 - Aachen, Germany
Duration: 26 Aug 201930 Aug 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume138
ISSN (Print)1868-8969

Conference

Conference44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019
Country/TerritoryGermany
CityAachen
Period26/08/1930/08/19

Keywords

  • Bisection
  • Chromatic coding
  • Fpt algorithm
  • Polynomial kernel
  • Semicomplete digraph
  • Splitters
  • Tournament

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