We study the classical NP-hard problems of finding maximum-size subsets from given sets of $k$ terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is $Ω(\log^{1/2-ε}{n})$, assuming NP$~\not\subseteq~$ZPTIME$(n^{\mathrm{poly}\log n})$. This constitutes a significant gap to the best known approximation upper bound of $O(\sqrt{n})$ due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an $O(1)$-approximation when edges (or nodes) may be used by $O(\frac{\log n}{\log\log n})$ paths.
In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number $r$ of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following.
* For MaxEDP, we give an $O(\sqrt{r}\cdot \log^{1.5}{kr})$-approximation algorithm. As $r\leq n$, up to logarithmic factors, our result strengthens the best known ratio $O(\sqrt{n})$ due to Chekuri et al.
* Further, we show how to route $Ω(\mathrm{OPT})$ pairs with congestion $O(\frac{\log{kr}}{\log\log{kr}})$, strengthening the bound obtained by the classic approach of Raghavan and Thompson.
* For MaxNDP, we give an algorithm that gives the optimal answer in time $(k+r)^{O(r)}\cdot n$. If $r$ is at most triple-exponential in $k$, this improves the best known algorithm for MaxNDP with parameter $k$, by Kawarabayashi and Wollan (STOC 2010).
We complement these positive results by proving that MaxEDP is NP-hard even for $r=1$, and MaxNDP is W$[1]$-hard for parameter $r$.
On Structural Parameterizations of the Edge Disjoint Paths Problem
在本文中，我们重新审视经典的边缘不相交路径（EDP）的问题，其中一个是一个无向图G和一组终端对P，并要求G是否包含一组成对的边缘不相交路径连接每个终端对P.我们的重点在于结构参数化的问题，允许有效的（多项式时间或FPT）算法。作为我们的第一个结果，我们回答了Fleszar，Mnich和Spoerhase（2016）中提出的一个开放性问题，通过证明如果输入图具有大小为1的反馈顶点集，则可以在多项式时间内解决该问题。我们还表明，EDP参数的树宽和最大程度的输入图是固定参数听话。已经开发了两种新的算法EDP使用结构上的限制输入图，然后我们把我们的注意力转向增强图，即，在每个端子对之间添加一条边之后从输入图获得的图。在对输入图的扩展中，即使树宽为2，EDP也是NP难的，Zhou et al.（2000）的结果表明，如果扩展图具有恒定的树宽，则EDP可以在非均匀多项式时间内求解;我们注意到，从那时起，这个结果对FPT算法的可能改进仍然是开放的。我们表明，这是非常不可能的，通过建立W[1]-硬度的问题参数化的树宽（甚至反馈顶点集）的增强图。最后，我们开发了一个FPT算法的EDP利用一个新的结构参数的增强图。
Average-Tree Phylogenetic Diversity Parameterized by Scanwidth and Invisibility
Delft University of Technology（代尔夫特理工大学）；Middlesex University（米德尔塞克斯大学）；ISEM, Université de Montpellier（蒙彼利埃大学生物演化与多样性研究所）；LIGM, CNRS, Univsersité Gustave Eiffel（巴黎-塞纳河东高等学院计算机科学与数字科学实验室）；Univsersité Gustave Eiffel（古斯塔夫·艾菲尔大学）
