New algorithms for maximum disjoint paths based on tree-likeness

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.