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arXiv 1107.1292cs.DM
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无子图和浅无子图的分离器定理及其应用

Separator Theorems for Minor-Free and Shallow Minor-Free Graphs with Applications

Christian Wulff-Nilsen

备注：
To appear at FOCS 2011
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摘要：

Alon，Seymour和Thomas推广了Lipton和Tarjan的平面分离器定理，证明了具有$n$顶点的$K_h$ -子自由图的分离器的大小至多为$h^{3/2}\sqrt n$ .他们给出了一个算法，给定一个图$G$与$m$边和$n$顶点，并给出一个整数$h\geq 1$，输出在$O(\sqrt{hn}m)$时间这样一个分隔符或$K_h$ -小$G$。Plotkin、Rao和Smith给出了一个$O(hm\sqrt{n\log n})$时间算法来寻找大小为$O(h\sqrt{n\log n})$的分隔符。Kawarabayashi和Reed改进了$h\sqrt n$的分隔符大小的界限，并给出了一个算法，该算法可以在$O(n^{1 + ε})$时间内找到这样一个分隔符，对于任何常数$ε> 0$，假设$h$是常数。该算法在运行时间上对$h$有极大的依赖性（$h$的一些功率塔，其高度本身是$h$的函数），使得即使对于小的$h$也不切实际。我们感兴趣的是一个小的多项式时间依赖$h$，我们展示了如何找到一个$O(h\sqrt{n\log n})$大小的分隔符或报告，$G$有一个$K_h$ -未成年人在$O(\poly(h)n^{5/4 + ε})$时间为任何常数$ε> 0$。我们还提出了第一个$O(\poly(h)n)$时间算法，找到一个常数$c < 1$的大小$O(n^c)$的分隔符。作为我们的结果的推论，我们得到了改进的最短路径和最大匹配算法。此外，对于整数$\ell$和$h$，我们给出了一个$O(m + n^{2 + ε}/\ell)$时间算法，该算法要么产生深度为$O(\ell\log n)$的$K_h$ -小调，要么产生大小至多为$O(n/\ell + \ell h^2\log n)$的分隔符。这改进了Plotkin、Rao和Smith在$m = Ω(n^{1 + ε})$时的浅层子算法。我们得到一个类似的运行时间的改进的近似算法的问题，找到一个最大的$K_h$ -未成年人在一个给定的图。

英文摘要：

Alon, Seymour, and Thomas generalized Lipton and Tarjan's planar separator theorem and showed that a $K_h$-minor free graph with $n$ vertices has a separator of size at most $h^{3/2}\sqrt n$. They gave an algorithm that, given a graph $G$ with $m$ edges and $n$ vertices and given an integer $h\geq 1$, outputs in $O(\sqrt{hn}m)$ time such a separator or a $K_h$-minor of $G$. Plotkin, Rao, and Smith gave an $O(hm\sqrt{n\log n})$ time algorithm to find a separator of size $O(h\sqrt{n\log n})$. Kawarabayashi and Reed improved the bound on the size of the separator to $h\sqrt n$ and gave an algorithm that finds such a separator in $O(n^{1 + ε})$ time for any constant $ε> 0$, assuming $h$ is constant. This algorithm has an extremely large dependency on $h$ in the running time (some power tower of $h$ whose height is itself a function of $h$), making it impractical even for small $h$. We are interested in a small polynomial time dependency on $h$ and we show how to find an $O(h\sqrt{n\log n})$-size separator or report that $G$ has a $K_h$-minor in $O(\poly(h)n^{5/4 + ε})$ time for any constant $ε> 0$. We also present the first $O(\poly(h)n)$ time algorithm to find a separator of size $O(n^c)$ for a constant $c < 1$. As corollaries of our results, we get improved algorithms for shortest paths and maximum matching. Furthermore, for integers $\ell$ and $h$, we give an $O(m + n^{2 + ε}/\ell)$ time algorithm that either produces a $K_h$-minor of depth $O(\ell\log n)$ or a separator of size at most $O(n/\ell + \ell h^2\log n)$. This improves the shallow minor algorithm of Plotkin, Rao, and Smith when $m = Ω(n^{1 + ε})$. We get a similar running time improvement for an approximation algorithm for the problem of finding a largest $K_h$-minor in a given graph.

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