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Abstract: We study the Unsplittable Flow Problem (UFP) on a line graph, 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 is contained in DTIME(2^polylog(n)). Our result requires a quasi-polynomial bound on all edge capacities and demands in the input instance.

Earlier results on this problem included a polynomial time (2+epsilon)-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.

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   Nikhil Bansal, Amit Chakrabarti, Amir Epstein, and Baruch Schieber, "A Quasi-PTAS for Unsplittable Flow on Line Graphs." Dartmouth Computer Science Technical Report TR2005-561, October 2005.

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