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

Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection 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...

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Bibliographic Details
Published in:Algorithmica Vol. 83; no. 6; pp. 1861 - 1884
Main Authors: Madathil, Jayakrishnan, Sharma, Roohani, Zehavi, Meirav
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2021
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Kernels
Mathematics of Computing
Polynomials
Theory of Computation
Bisection
Polynomial kernel
Splitters
FPT Algorithm
Chromatic coding
Semicomplete digraph
Tournament
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection 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 M inimum D irected B isection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that M inimum D irected B isection 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 M inimum D irected B isection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-021-00806-x