New algorithms for maximum disjoint paths based on tree-likeness

We study the classical  $$ -hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is  $$ , assuming  $$ . This constitutes a significant gap to the best known approximation upper bound of  $$ 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  $$ -approximation when edges (or nodes) may be used by  $$ paths. In this paper, we strengthen the fundamental results above. 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 results:For MaxEDP, we give an  $$ -approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio  $$ due to Chekuri et al., as  $$ .Further, we show how to route  $$ pairs with congestion bounded by  $$ , strengthening the bound obtained by the classic approach of Raghavan and Thompson.For MaxNDP, we give an algorithm that gives the optimal answer in time  $$ . This is a substantial improvement on the run time of  $$ , which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is  $$ -hard even for  $$ , and MaxNDP is  $$ -hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless  $$ and that our approximability results are relevant even for very small constant values of r.