基于动态近似距离预言机的浅无子图快速分离器
Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles
摘要：
Plotkin，Rao，and Smith（SODA'97）证明了任何具有$m$边和$n$顶点的图，如果不将$K_h$作为深度$O(\ell\log n)$ -minor，则具有大小为$O(n/\ell + \ell h^2\log n)$的分隔符，并且可以在$O(mn/\ell)$时间中找到这样的分隔符。随后给出了任何恒定$ε> 0$的$O(m + n^{2+ε}/\ell)$的时间界限（W.，FOCS'11），这是对非稀疏图的改进。我们给出了三个新的算法。第一个具有相同的分隔符大小和运行时间$O(\mbox{poly}(h)\ell m^{1+ε})$。这对于小型$h$和$\ell$是一个显著的改进。如果$\ell = Ω(n^{ε'})$对于任意小的常数$ε' > 0$，我们得到$O(\mbox{poly}(h)\ell n^{1+ε})$的时间界。第二种算法实现了相同的分隔符大小（对$h$的多项式依赖性稍大）和运行时间$O(\mbox{poly}(h)(\sqrt\ell n^{1+ε} + n^{2+ε}/\ell^{3/2}))$，当$\ell = Ω(n^{ε'})$。我们的第三个算法有运行时间$O(\mbox{poly}(h)\sqrt\ell n^{1+ε})$时，$\ell = Ω(n^{ε'})$。它找到一个大小为$O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n)$的分隔符，该分隔符在$h$固定且$\ell = \tilde O(n^{1/4})$。在获得我们的结果的主要工具是一个新的应用递减近似距离预言的Roditty和Zwick。
英文摘要：
Plotkin, Rao, and Smith (SODA'97) showed that any graph with $m$ edges and $n$ vertices that excludes $K_h$ as a depth $O(\ell\log n)$-minor has a separator of size $O(n/\ell + \ell h^2\log n)$ and that such a separator can be found in $O(mn/\ell)$ time. A time bound of $O(m + n^{2+ε}/\ell)$ for any constant $ε> 0$ was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time $O(\mbox{poly}(h)\ell m^{1+ε})$. This is a significant improvement for small $h$ and $\ell$. If $\ell = Ω(n^{ε'})$ for an arbitrarily small chosen constant $ε' > 0$, we get a time bound of $O(\mbox{poly}(h)\ell n^{1+ε})$. The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on $h$) and running time $O(\mbox{poly}(h)(\sqrt\ell n^{1+ε} + n^{2+ε}/\ell^{3/2}))$ when $\ell = Ω(n^{ε'})$. Our third algorithm has running time $O(\mbox{poly}(h)\sqrt\ell n^{1+ε})$ when $\ell = Ω(n^{ε'})$. It finds a separator of size $O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n)$ which is no worse than previous bounds when $h$ is fixed and $\ell = \tilde O(n^{1/4})$. A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
