New algorithms for maximum disjoint paths based on tree-likeness

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 2Ω(logn), 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 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 O(rlog(kr))-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 (kr) / log log (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) ^{O ( r )}· n. This is a substantial improvement on the run time of 2 ^kr ^{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.