上层为树的两层图绘制：顶点分裂与固定参数算法
Two-Layer Drawings with a Tree on Top: Vertex Splits and Fixed-Parameter Algorithms
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中文总结 AI 辅助
研究两层图绘制中顶点分裂问题的扩展，其中一层顶点顺序受辅助树\(T\)约束。贡献为给出关于分裂数\(k\)及\(T\)最大度的固定参数算法，还改进经典无约束版本算法，并通过实现验证算法在实践中的良好表现。
中文摘要 AI 辅助
二分图的两层图绘制将每部分顶点置于两条平行线之一上，并将边绘制成直线连接。传统上，优化目标是找到一层或两层上的顶点排列，以最小化诱导的边交叉数。此问题是NP难的，且交叉最小的解可能仍有许多交叉。近来，通过顶点分裂（即用两个或更多副本替换原始顶点并在它们之间分配邻接关系）来消除所有交叉这一正交优化目标受到更多关注。本文研究两层顶点分裂问题的自然扩展，其中一层上的顶点顺序受给定辅助树\(T\)约束，该问题受人类参考图谱中解剖层次可视化等应用的推动。我们研究此问题的参数化复杂度并获得两个主要贡献：（1）关于分裂数\(k\)的固定参数算法；（2）关于\(T\)的最大度的ETH紧单指数固定参数算法。此外，基于后一结果，我们为该问题的经典无约束版本获得了一个ETH紧单指数算法，改进了之前的\(O^*(2^{k\cdot \log k})\)算法。最后，我们实现了算法并表明它在实践中表现良好。
英文摘要
Two-layer drawings of bipartite graphs place the vertices of each part on one of two parallel lines and draw the edges as straight-line links. Traditionally, the optimization goal is to find vertex permutations on one or both layers that minimize the induced number of edge crossings. This problem is NP-hard, and crossing-minimal solutions may still contain many crossings. Recently, there has been growing interest in an orthogonal optimization goal, namely removing all crossings by vertex splitting, i.e., replacing original vertices by two or more copies and distributing the adjacencies among them. In this paper, we study a natural extension of the two-layer vertex splitting problem in which the vertex order on one layer is constrained by a given auxiliary tree $T$, motivated by applications such as the visualization of anatomical hierarchies in the Human Reference Atlas. We investigate the parameterized complexity of this problem and obtain two main contributions: (1) a fixed-parameter algorithm with respect to the number $k$ of splits, and (2) an ETH-tight single-exponential fixed-parameter algorithm with respect to the maximum degree of $T$. Moreover, we build on the latter result to obtain an ETH-tight single-exponential algorithm for the classical unconstrained version of the problem, improving upon the previous $O^*(2^{k\cdot \log k})$ algorithms. Finally, we also implement our algorithm and show that it performs well in practice.
