TY - GEN

T1 - A sub-exponential FPT algorithm and a polynomial kernel for minimum directed bisection on semicomplete digraphs

AU - Madathil, Jayakrishnan

AU - Sharma, Roohani

AU - Zehavi, Meirav

N1 - Funding Information:
Supported by ISF grant no.
Funding Information:
Meirav Zehavi: Supported by ISF grant no. 1176/18. We thank Daniel Lokshtanov and Saket Saurabh for insightful discussions on bisection in semicomplete digraphs.
Publisher Copyright:
© Jayakrishnan Madathil, Roohani Sharma, and Meirav Zehavi.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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.

AB - 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.

KW - Bisection

KW - Chromatic coding

KW - Fpt algorithm

KW - Polynomial kernel

KW - Semicomplete digraph

KW - Splitters

KW - Tournament

UR - http://www.scopus.com/inward/record.url?scp=85071780429&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2019.28

DO - 10.4230/LIPIcs.MFCS.2019.28

M3 - Conference contribution

AN - SCOPUS:85071780429

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019

A2 - Katoen, Joost-Pieter

A2 - Heggernes, Pinar

A2 - Rossmanith, Peter

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019

Y2 - 26 August 2019 through 30 August 2019

ER -