Mim-Width is paraNP-complete

TL;DR

The paper establishes paraNP-completeness for computing Mim-width, Sim-width, One-Sided Mim-width, and their linear variants by a three-stage reduction from 4-Occ Not-All-Equal 3-Sat. It introduces Linear Degree Balancing and Tree Degree Balancing to translate satisfiability into balancing orders, then encodes those restrictions into Mim-/Sim-Balancing problems, and finally把 those into Mim-/Sim-Width through carefully designed gadgets and constructions. The result provides a fixed, constant gap (e.g., $1211$ vs $1216$) that implies there is no XP algorithm under ETH for computing these widths, and it presents new structural techniques that could apply to other width notions defined by branch decompositions. Overall, the work unifies the hardness of several width parameters and clarifies the parameterized complexity landscape for these graph-width measures.

Abstract

We show that it is NP-hard to distinguish graphs of linear mim-width at most 1211 from graphs of sim-width at least 1216. This implies that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear counterparts are all paraNP-complete, i.e., NP-complete to compute even when upper bounded by a constant.

Figures (9)

• Figure 1: Visual summary of our reduction, split into its three steps.
• Figure 2: Illustration of a $(\tau,\gamma)$-bottleneck.
• Figure 3: Bottleneck sequence $B(S_1,S_2,S_3)$. Vertices of $S_1 \cup \{s_1\}, S_2 \cup \{s_2\}, S_3 \cup \{s_3\}$ are in red, green, and blue, respectively. As in \ref{['fig:bottleneck']}, every edge with an unspecified weight get one in the discrete interval $[\gamma+1,\tau-\gamma-1]$.
• Figure 4: Illustration of $(H, \omega)$. Centered at the top is the bottleneck sequence $B(T, C, F)$. The vertices of $X$ are in purple (left), and the vertices of $Y$ are in yellow (right). The edges incident to the variable vertices that are drawn in blue, green, red all have weight $\lambda$. Not to overburden the figure, we have only drew some edges of the construction. Only one edge of the matching between $X$ and $Y$ is depicted, and the paddings of $\overline{v_x}$ and of $\overline{t_2}$ are (partially) represented (weight-1 edges toward $X$). The clause corresponding to the bottommost vertex of $C$ contains $x, y$ and some other variable (not shown), while that of the second bottommost vertex of $C$ contains $z$ (and two other variables). The roots of bottlenecks are in gray. The leftmost and rightmost gray vertices are the only two vertices of weight less than $\tau+\gamma+1$ (namely $\tau$).
• Figure 5: Adjacencies between $S(u)$ and $S(v)$. In this example, $u$ has four neighbors $v, v_1, v_2, v_3$, and $v$ has three neighbors $u, v_3, v_4$. The matching edges are in blue, the dummy edges are in black (edges between two boxes represent bicliques). Notice the non-edges between $I(u,v_3)$ and $I(v,v_3)$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Definition 6
• Lemma 7
• Definition 8
• Lemma 9
• Lemma 10
• Lemma 11