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A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs

Authors Jayakrishnan Madathil, Roohani Sharma, Meirav Zehavi



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Jayakrishnan Madathil
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Roohani Sharma
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav Zehavi
  • Ben-Gurion University, Beer-Sheva, Israel

Acknowledgements

We thank Daniel Lokshtanov and Saket Saurabh for insightful discussions on bisection in semicomplete digraphs.

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Jayakrishnan Madathil, Roohani Sharma, and Meirav Zehavi. A Sub-Exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.28

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 2^{O(sqrt{k} log k)}n^{O(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,k^2)-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • bisection
  • semicomplete digraph
  • tournament
  • fpt algorithm
  • chromatic coding
  • polynomial kernel
  • splitters

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