We study the problem of routing on disjoint paths in bounded treewidth graphs
with both edge and node capacities. The input consists of a capacitated graph
G and a collection of k source-destination pairs M={(s1​,t1​),…,(sk​,tk​)}. The goal is to maximize the number of pairs that
can be routed subject to the capacities in the graph. A routing of a subset
M′ of the pairs is a collection P of paths such that,
for each pair (si​,ti​)∈M′, there is a path in P
connecting si​ to ti​. In the Maximum Edge Disjoint Paths (MaxEDP) problem,
the graph G has capacities cap(e) on the edges and a routing
P is feasible if each edge e is in at most cap(e) of
the paths of P. The Maximum Node Disjoint Paths (MaxNDP) problem is
the node-capacitated counterpart of MaxEDP.
In this paper we obtain an O(r3) approximation for MaxEDP on graphs of
treewidth at most r and a matching approximation for MaxNDP on graphs of
pathwidth at most r. Our results build on and significantly improve the work
by Chekuri et al. [ICALP 2013] who obtained an O(râ‹…3r) approximation
for MaxEDP