A quasi-PTAS for unsplittable flow on line graphs

We study the Unsplittable Flow Problem (UFP) on line graphs and cycles, focusing on the long-standing open question of whether the problem is APX-hard. We describe a deterministic quasi-polynomial time approximation scheme for UFP on line graphs, thereby ruling out an APX-hardness result, unless NP ¿ DTIME(2polylog(n)). Our result requires a quasi-polynomial bound on all edge capacities and demands in the input instance. We extend this result to undirected cycle graphs.Earlier results on this problem included a polynomial time (2+e)-approximation under the assumption that no demand exceeds any edge capacity (the no-bottleneck assumption ) and a super-constant integrality gap if this assumption did not hold. Unlike most earlier work on UFP, our results do not require a no-bottleneck assumption