Approximability of the Unsplittable Flow Problem on Trees

We consider the approximability of the Unsplittable Flow Problem (UFP) on tree graphs, and give a deterministic quasi-polynomial time approximation scheme for the problem when the number of leaves in the tree graph is at most poly-logarithmic in $n$ (the number of demands), and when all edge capacities and resource requirements are suitably bounded. Our algorithm generalizes a recent technique that obtained the first such approximation scheme for line graphs. Our results show that the problem is not APX-hard for such graphs unless NP \subseteq DTIME(2^{polylog(n)}). Further, a reduction from the Demand Matching Problem shows that UFP is APX-hard when the number of leaves is Omega(n^\epsilon) for any constant \epsilon \u3e 0. Together, the two results give a nearly tight characterization of the approximability of the problem on tree graphs in terms of the number of leaves, and show the structure of the graph that results in hardness of approximation