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, ), 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, ) 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: class="unordered" style="list-style-type:disc">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 id="IEq12">. This is a substantial improvement on the run time of id="IEq13">, which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that class="small-caps">MaxEDP is id="IEq14">-hard even for id="IEq15">, and class="small-caps">MaxNDP is id="IEq16">-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless id="IEq17"> and that our approximability results are relevant even for very small constant values of r.
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