## 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 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 admits a sub-exponential time fixed-parameter tractable 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, which, to the best of our knowledge, have never been used before in the design of kernels. We also prove that Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.

Original language | English |
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Pages (from-to) | 1861-1884 |

Number of pages | 24 |

Journal | Algorithmica |

Volume | 83 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jun 2021 |

## Keywords

- Bisection
- Chromatic coding
- FPT Algorithm
- Polynomial kernel
- Semicomplete digraph
- Splitters
- Tournament

## ASJC Scopus subject areas

- Computer Science (all)
- Computer Science Applications
- Applied Mathematics