Fixed-parameter tractability of DIRECTED MULTICUT with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

Abstract

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 (calledterminals) (s1, t1), (s2, t2), (s3, t3), and an integer k and we want to find a set of at most k non-terminalvertices 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 ofthe two-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of thefour-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the techniqueof 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 problemcan be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. Welook 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 thematrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the gridminor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced.