Abstract
We show fixed-parameter tractability of the Directed Multicut problem with
three terminal pairs (with a randomized algorithm). This problem, given a
directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$,
$(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most
$k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all
$s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this
case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved
fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and
Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-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, Wahlstr\"{o}m, 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,
Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,
Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, 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.
