Abstract
The Unsplittable Flow Problem on a Path (UFPP) is a core problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a source and a destination vertex on the path. The goal is to compute a subset of tasks of maximum profit that does not violate the edge capacities.
In practical applications generic approaches such as integer programming (IP) methods are desirable. Unfortunately, no IP-formulation is known for the problem whose LP-relaxation has an integrality gap that is provably constant. For the unweighted case, we show that adding a few constraints to the standard LP of the problem is sufficient to make the integrality gap drop from Ω(n) to O(1). This positively answers an open question in [Chekuri et al., APPROX 2009].
For the general (weighted) case, we present an extended formulation with integrality gap bounded by 7 + ε. This matches the best known approximation factor for the problem [Bonsma et al., FOCS 2011]. This result exploits crucially a technique for embedding dynamic programs into linear programs. We believe that this method could be useful to strengthen LP-formulations for other problems as well and might eventually speed up computations due to stronger problem formulations.
Partially supported by EU FP7 Project N. 255403 SNAPS, by the ERC Starting Grant NEWNET 279352, by the ERC Starting Grant PAAl 259515, and by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).
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Anagnostopoulos, A., Grandoni, F., Leonardi, S., Wiese, A. (2013). Constant Integrality Gap LP Formulations of Unsplittable Flow on a Path. In: Goemans, M., Correa, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2013. Lecture Notes in Computer Science, vol 7801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36694-9_3
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