We study the classical $${\mathsf {NP}}$$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/MaxNDP is currently not well understood; the best known lower bound is $${2^{\varOmega (\sqrt{\log n})}}$$2Ω(logn), assuming $${\mathsf {NP}\not \subseteq \mathsf {DTIME}(n^{\mathcal {O}(\log n)})}$$NP⊈DTIME(nO(logn)). This constitutes a significant gap to the best known approximation upper bound of $${\mathcal {O}(\sqrt{n})}$$O(n) due to Chekuri et al. (Theory Comput 2:137–146, 2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica 7(4):365–374, 1987) introduce the technique of randomized rounding for LPs; their technique gives an $${\mathcal {O}(1)}$$O(1)-approximation when edges (or nodes) may be used by $${\mathcal {O}\left( \log n/\log \log n\right) }$$Ologn/loglogn paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set numberr of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:For MaxEDP, we give an $${\mathcal {O}(\sqrt{r} \log ({k}r))}$$O(rlog(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio $${\mathcal {O}(\sqrt{n})}$$O(n) due to Chekuri et al., as $${r\le n}$$r≤n.Further, we show how to route $${\varOmega ({\text {OPT}}^{*})}$$Ω(OPT∗) pairs with congestion bounded by $${\mathcal {O}(\log (kr)/\log \log (kr))}$$O(log(kr)/loglog(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 $${(k+r)^{\mathcal {O}(r)}\cdot n}$$(k+r)O(r)·n. This is a substantial improvement on the run time of $${2^kr^{\mathcal {O}(r)}\cdot n}$$2krO(r)·n, which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is $${\mathsf {NP}}$$NP-hard even for $${r=1}$$r=1, and MaxNDP is $${\mathsf {W}[1]}$$W[1]-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless $${\mathsf {FPT}= \mathsf {W}[1]}$$FPT=W[1] and that our approximability results are relevant even for very small constant values of r.

中文翻译：

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基于树状的最大不相交路径的新算法

我们研究了经典的 $${\mathsf {NP}}$$NP-hard 问题，即从给定的 k 个终端对集合中找到最大大小的子集，这些终端对可以通过边不相交路径 (MaxEDP) 或节点不相交路径 ( MaxNDP）在给定的图中。MaxEDP/MaxNDP的逼近性目前还不是很清楚；最知名的下界是 $${2^{\varOmega (\sqrt{\log n})}}$$2Ω(logn)，假设 $${\mathsf {NP}\not \subseteq \mathsf {DTIME} (n^{\mathcal {O}(\log n)})}$$NP⊈DTIME(nO(logn))。由于 Chekuri 等人，这与 $${\mathcal {O}(\sqrt{n})}$$O(n) 的最知名近似上限存在显着差距。（Theory Comput 2:137–146, 2006），缩小这一差距是目前逼近算法中的一大悬而未决的问题。在他们的开创性论文中，Raghavan 和 Thompson (Combinatorica 7(4):365–374, 1987) 介绍了 LP 的随机舍入技术；当 $${\mathcal {O}\left( \log n/\ log \log n\right) }$$Ologn/loglogn 路径。在本文中，我们加强了上述基本结果。我们提供了根据图的反馈顶点集编号制定的新界限，它测量其到森林的顶点删除距离。特别是，我们得到以下结果：对于 MaxEDP，我们给出一个 $${\mathcal {O}(\sqrt{r} \log ({k}r))}$$O(rlog(kr))-近似算法。由于 Chekuri 等人，我们的结果加强了最知名的比率 $${\mathcal {O}(\sqrt{n})}$$O(n)，因为 $${r\le n }$$r≤n。进一步，我们展示了如何路由 $${\varOmega ({\text {OPT}}^{*})}$$Ω(OPT∗) 对，拥塞限制为 $${\mathcal {O}(\log (kr) /\log \log (kr))}$$O(log(kr)/loglog(kr))，加强由Raghavan和Thompson的经典方法获得的界限。对于MaxNDP，我们给出了一个给出最优答案的算法在时间 $${(k+r)^{\mathcal {O}(r)}\cdot n}$$(k+r)O(r)·n。这是对 $${2^kr^{\mathcal {O}(r)}\cdot n}$$2krO(r)·n 的运行时间的实质性改进，可以通过 Scheffler 的算法获得。我们通过证明即使对于 $${r=1}$$r=1，MaxEDP 也是 $${\mathsf {NP}}$$NP-hard 来补充这些积极的结果，并且 MaxNDP 是 $${\mathsf {W} [1]}$$W[1]-当 r 为参数时是困难的。