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

TL;DR

This work resolves the parameterized complexity of Directed Multicut with three terminal pairs by giving a randomized fixed-parameter tractable algorithm parameterized by the cutset size $k$. The authors combine flow-augmentation with a CSP encoding and a twin-width framework to obtain an FPT procedure: after shadow-removal and sampling steps, the remaining constraints have bounded grid rank, enabling FO-model checking on a graph of bounded twin-width. The reduction to a Permutation CSP with a small number of variables and a controlled permutation constraint structure is central, with an irrelevant-vertex rule ensuring bounded grid minors. The paper also establishes a hardness barrier for Weighted Directed Multicut with two terminals, underscoring the boundary of tractability in directed separation problems and situating the results within the broader twin-width/FO-model checking paradigm.

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 Wahlström 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, 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 Mendes, 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.

Figures (4)

• Figure 1: This flowchart gives an overview on the structure of the algorithm and where the results from the other sections are used.
• Figure 2: The downwards-closed constraints introduced into the Permutation CSP instance $\mathcal{C}_1$. All identifiers $\leq, \leq', R, P$ to be understood with an index $^i$.
• Figure 3: An illustration of the construction of the paths $X^{i,j}$, $Y^{i,j}$, $X^{j,i}$, $Y^{j,i}$ and the grid $P^{i,j}$, where $i<j$ and $n=5$. The brown vertices whose labels have subscript $a$ have weight $Ma$. The green vertices whose labels have subscript $a$ have weight $M(n+1-a)$. The black vertices have weight $1$ and the white vertices are undeletable. The red, and the blue, highlighted paths are the $s_1t_1$-path, and $s_2t_2$-path respectively, that survive after deleting the solution vertices from $X^{i,j}$, $Y^{i,j}$, $X^{j,i}$ and $Y^{j,i}$. The unique common intersection point of the two highlighted paths is $p^{i,j}_{a,b}$, which is picked by the solution. The vertices picked by the solution are encircled in red.
• Figure 4: The paths $X^{i,j}$, $Z_i$ and $Y^{i,j}$ are shown. If the index of a vertex is $a$, then its weight is $M$ if it the vertex is blue, $Ma$ if the vertex is brown, $M(n+1-a)$ if the vertex is green and $W+1$ otherwise. This gadget together with the definition of the target weight $W$ of the solution, ensures that a solution picks exactly one vertex from each of the three paths. Moreover, the three vertices chosen from these three paths have the same index.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Chitnis et al. dsfvs, Lemma $3.11$
• Theorem 2.2
• Theorem 2.3: Bonnet et al. BonnetGMSTT22
• Lemma 2.4: Bonnet et al. BonnetCKKLT22
• Theorem 2.5
• Definition 2.6
• Theorem 2.7