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We study the classical   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   Ω  (    log  n    )    , assuming    NP  ⊈  DTIME  (   n   O  (  log  n  )    )   . This constitutes a significant gap to the best known approximation upper bound of    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    O  (  1  )   -approximation when edges (or nodes) may be used by    O   log  n  /  log  log  n    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    O  (   r   log   (  k  r  )   )   -approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio    O  (   n   )   due to Chekuri et al., as    r  ≤  n   .Further, we show how to route    Ω  (    OPT∗    )   pairs with congestion bounded by    O  (  log  (  k  r  )  /  log  log  (  k  r  )  )   , 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  )    O  (  r  )    ·  n   . This is a substantial improvement on the run time of     2  k    r   O  (  r  )    ·  n   , which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is   NP  -hard even for    r  =  1   , and MaxNDP is    W  [  1  ]   -hard when  r is the parameter. This shows that neither problem is fixed-parameter tractable in  r unless    FPT  =  W  [  1  ]   and that our approximability results are relevant even for very small constant values of  r .

New algorithms for maximum disjoint paths based on tree-likeness