Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization

TL;DR

This work resolves the subexponential fixed-parameter complexity of 2-Layer Crossing Minimization and, more generally, delineates tractability boundaries for h-Layer Crossing Minimization. It introduces a drawing-based separator technique that drives a divide-and-conquer approach, enabling subexponential time algorithms for h=2 ($2^{O(\, oot k \,)}$-like in the parameter) and h=3 ($2^{O(k^{2/3} ext{log} k)}$), while proving ETH-based nonexistence of $2^{o(k/ ext{log} k)}$-time algorithms for h≥5. A polynomial kernel is achieved for h=3 (size O$(k^{8})$) and a dichotomy rules out polynomial kernels for h≥4, underpinning the kernelization aspect of the subexponential framework. The methodology hinges on separators aligned with the drawing, careful guessing of interfaces, and a sequence of reduction rules that preserve solvability while bounding instance size. The results collectively advance understanding of when layered crossing minimization is feasible in subexponential time and what kernel sizes are possible, with clear open problems for the remaining 4-layer case and tighter exponent bounds.

Abstract

The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of \textsc{2-Layer Crossing Minimization}. In this problem, the input is an $n$-vertex graph $G$ whose vertices are partitioned into two independent sets $V_1$ and $V_2$, and a non-negative integer $k$. The question is whether $G$ admits a 2-layered drawing with at most $k$ crossings, where each $V_i$ lies on a distinct line parallel to the $x$-axis, and all edges are straight lines. We resolve this open question by giving the first subexponential fixed-parameter algorithm for this problem, running in time $2^{O(\sqrt{k}\log k)} + n \cdot k^{O(1)}$. We then ask whether the subexponential phenomenon extends beyond two layers. In the general $h$-Layer Crossing Minimization problem, the vertex set is partitioned into $h$ independent sets $V_1, \ldots, V_h$, and the goal is to decide whether an $h$-layered drawing with at most $k$ crossings exists. We present a subexponential FPT algorithm for three layers with running time $2^{O(k^{2/3}\log k)} + n \cdot k^{O(1)}$ for $h = 3$ layers. In contrast, we show that for all $h \ge 5$, no algorithm with running time $2^{o(k/\log k)} \cdot n^{O(1)}$ exists unless the Exponential-Time Hypothesis fails. Finally, we address polynomial kernelization. While a polynomial kernel was already known for $h=2$, we design a new polynomial kernel for $h=3$. These kernels are essential ingredients in our subexponential algorithms. Finally, we rule out polynomial kernels for all $h \ge 4$ unless the polynomial hierarchy collapses.

Figures (37)

• Figure 1: Example of $3$-layered drawings of the same graph with five and two crossings.
• Figure 2: Illustration of a separator. The edges ${\sf lsep}$ and ${\sf rsep}$ are shown in blue, and their endpoints are shown in purple. It is possible that the endpoints on $V_1$ (or $V_2$) are the same. The edges between ${\sf lsep}$ and ${\sf rsep}$ are shown in brown, and the edges that cross ${\sf lsep}$ or ${\sf rsep}$ are shown in olive. Vertices between the endpoints of ${\sf lsep}$ and ${\sf rsep}$ are shown in dark green (denoted ${\sf Mid}_1$ for $V_1$) or light pink (denoted ${\sf Mid}_2$ for $V_2$).
• Figure 3: Illustration of the problematic components described in the overview. Here, the endpoint on $V_2$ of the two separator edges (colored blue) is shared. The edges of the three "leaf components" are denoted by dashed black lines. Each of the other five problematic components is denoted using a different color: four of them (colored purple, green, orange and red) have a single edge each, and the fifth (colored golden) has three edges---the edges connecting the component to the separator vertex on top are not part of the component, but are drawn using the same color for clarity.
• Figure 4: Illustration of a separator. The four edges of the separator are shown in blue, and their endpoints are shown in purple. It is possible that the some of the endpoints of the separator edges are the same. The edges between the left are right parts of the separator whose number is guaranteed to be upper-bounded (specifically, here, those between $V_1$ and $V_2$) are shown in brown, and the edges that cross the separator edges are shown in olive. Vertices on $V_2$ between the endpoints of the two right separator edges and which are incident to edges above them (these edges are not illustrated) are shown in dark green (denoted GapRight), and vertices on $V_2$ between the endpoints of the two left separator edges and which are incident to edges above them are shown in orange (denoted GapLeft).
• Figure 5: Illustration of components ${\sf LeftmostLC}, {\sf RightmostLC}$ and star vertices. Left-star vertices (shown in green) have at least one incident edge crossing $vu_l$, and right-star (purple) have at least one incident edge crossing $vu_r$. Note that a vertex can be both left- and right-star vertex, e.g., $p$.

Theorems & Definitions (174)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 1: JungerLM98
• proof
• Proposition 2: bla2024constrained
• Proposition 3: KobayashiMNT14
• proof : Proof of \ref{['prop:kern-two']}: Sketch of the running time evaluation