New algorithms for maximum disjoint paths based on tree-likeness

We study the classical [Formula: see text] -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 [Formula: see text] , assuming [Formula: see text] . This constitutes a significant gap to the best known approximation upper bound of [Formula: see text] 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 [Formula: see text] -approximation when edges (or nodes) may be used by [Formula: see text] paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r For MaxEDP, we give an [Formula: see text] -approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio [Formula: see text] due to Chekuri et al., as [Formula: see text] . Further, we show how to route [Formula: see text] pairs with congestion bounded by [Formula: see text] , 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 [Formula: see text] . This is a substantial improvement on the run time of [Formula: see text] , which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is [Formula: see text] -hard even for [Formula: see text] , and MaxNDP is [Formula: see text] -hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless [Formula: see text] and that our approximability results are relevant even for very small constant values of r.