TY - GEN

T1 - Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset

T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

AU - Hatzel, Meike

AU - Jaffke, Lars

AU - Lima, Paloma T.

AU - Masařík, Tomáš

AU - Pilipczuk, Marcin

AU - Sharma, Roohani

AU - Sorge, Manuel

N1 - Publisher Copyright:
Copyright © 2023.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). In this problem we are given a directed graph G, three pairs of vertices (called terminals) (s1, t1), (s2, t2), (s3, t3), and an integer k and we want to find a set of at most k non-terminal vertices in G that intersect all s1t1-paths, all s2t2-paths, and all s3t3-paths. The parameterized complexity of this problem has been open since Chitnis, Hajiaghayi, and Marx proved fixed-parameter tractability of the two-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the four-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant - not contained in the sought solution - and thus reduced.

AB - We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). In this problem we are given a directed graph G, three pairs of vertices (called terminals) (s1, t1), (s2, t2), (s3, t3), and an integer k and we want to find a set of at most k non-terminal vertices in G that intersect all s1t1-paths, all s2t2-paths, and all s3t3-paths. The parameterized complexity of this problem has been open since Chitnis, Hajiaghayi, and Marx proved fixed-parameter tractability of the two-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the four-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant - not contained in the sought solution - and thus reduced.

UR - http://www.scopus.com/inward/record.url?scp=85170067487&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85170067487

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 3229

EP - 3244

BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

PB - Association for Computing Machinery

Y2 - 22 January 2023 through 25 January 2023

ER -