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

2025-10-15

AI Summary

This paper investigates the parameterized complexity of minimizing edge crossings in layered graph drawing. Specifically, it addresses the $k$-Crossing Number problem on $2$- and $h$-layered graphs—i.e., deciding whether a layered drawing with at most $k$ crossings exists. The authors present the first subexponential fixed-parameter tractable (FPT) algorithms: $2^{O(sqrt{k} log k)}$ time for $2$ layers and $2^{O(k^{2/3} log k)}$ for $3$ layers; they further prove that no subexponential FPT algorithm exists for $h geq 5$, assuming the Exponential Time Hypothesis (ETH). This resolves a decade-old open problem. Moreover, they establish tight complexity bounds: for $h geq 4$, the problem admits no polynomial kernel, thereby refuting the existence of polynomial kernelization. The approach integrates advanced parameterized algorithm design, fine-grained reductions, and ETH-based lower bound analysis, systematically characterizing the stepwise increase in computational hardness as the number of layers grows.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of extsc{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.

Problem

• Resolving subexponential parameterized complexity for 2-layer crossing minimization
• Extending subexponential algorithms to three-layer graph drawings
• Establishing kernelization limits for layered crossing minimization problems

Innovation

• Subexponential FPT algorithm for 2-layer crossing minimization
• Subexponential FPT algorithm for 3-layer crossing minimization
• Polynomial kernelization for 2-layer and 3-layer cases