关于几何生成子图的（有向）宽度参数
On (Directed) Width-Parameters of Geometric Spanners
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中文总结 AI 辅助
本文研究几何生成子图的宽度参数，证明路径宽度等参数下存在渐近最优的稀疏生成子图，并指出树深度情形下膨胀比不可有界，且最小膨胀比问题在树深度3时NP-hard，给出XP算法。
中文摘要 AI 辅助
为了加速几何图上的算法，通常用保持某些几何性质的图来近似完全欧几里得图。欧几里得空间中点集 $P$ 的一个（有向）$t$-生成子图 $G$ 是一个（有向）图，使得对于任意一对点，$G$ 中的最短路径长度至多是这两点欧几里得距离的 $t$ 倍。本文研究受某些图参数限制的 $t$-生成子图。设 $\kappa$ 为一个图参数。我们证明，对于路径宽度、分支宽度和割宽度，存在 $P$ 上的一个 $\mathcal{O}(n/k^{d/(d-1)})$-生成子图 $G$ 满足 $\kappa(G)=k$，并且这在渐近最坏情况下是最优的。在 $\mathbb{R}^2$ 中，我们证明了对于团宽度或秩宽度为 $k$ 的平面图，同样的界成立。相反，对于树深度，我们证明存在一些点集，其膨胀比无法有界。因此，我们研究计算具有树深度 $k$ 和最小膨胀比的生成子图。我们证明，即使对于树深度 $3$，该问题在任意小于 $\sqrt{2}$ 的因子内近似都是 NP-困难的，并给出一个 XP 算法，对于给定的树深度 $k$，计算膨胀比至多为 $2t^*$ 的图，其中 $t^*$ 是最小膨胀比。我们进一步将这些结果扩展到有向情形，对于有向树宽度、有向路径宽度或 DAG 宽度 $\kappa$，得到满足 $\kappa(G)=k$ 的有向 $\mathcal{O}(n/k^{d/(d-1)})$-生成子图 $G$，并证明在有向情形下这也是渐近最坏情况最优的。
英文摘要
To speed up algorithms on geometric graphs, it is common to approximate the complete Euclidean graph while maintaining certain geometric properties. A (directed) $t$-spanner $G$ for a point set $P$ in the Euclidean space is a (directed) graph such that for every pair of points, the shortest path in $G$ is at most a factor $t$ longer than the Euclidean distance between those points. In this paper, we investigate $t$-spanners that are bounded by certain graph parameters. Let $κ$ be a graph parameter. We show that for path-width, branch-width and cut-width there is an $\mathcal{O}(n/k^{d/(d-1)})$-spanner $G$ on $P$ with $κ(G)=k$ and that this is asymptotically worst-case optimal. In $\mathbb{R}^2$ we show the same bounds for planar graphs of clique-width or rank-width $k$. In contrast, for tree-depth, we show that there are sets of points for which the dilation cannot be bounded. Therefore, we investigate computing a spanner with tree-depth $k$ and minimum dilation. We show that already for tree-depth $3$ this problem is NP-hard to approximate within any factor strictly less than $\sqrt{2}$, and present an XP-algorithm to compute for a given tree-depth $k$ a graph with dilation at most $2t^*$, where $t^*$ is the minimum dilation. We further extend these results to obtain directed $\mathcal{O}(n/k^{d/(d-1)})$-spanners $G$ with $κ(G)=k$ for $κ$ being directed tree-width, directed path-width or DAG-width and show that also in the directed case, this is asymptotically worst-case optimal.
发表机构
- TU Dortmund（多特蒙德工业大学）
- TU Eindhoven（埃因霍温理工大学）
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