A randomized rounding approach to a k-splittable multicommodity flow problem with lower path flow bounds affording solution quality guarantees

Abstract

In 2005, Baier et al. introduced a “k-splittable” variant of the multicommodity flow (MCF) problem in which a given commodity was allowed to flow through a number of paths limited by a small integer. This problem enables a better use of the link capacities than the classical Kleinberg’s unsplittable MCF problem while not overloading the used devices and protocols with a large number of paths. We solve a minimum-congestion k-splittable MCF problem coming from a practice of managing an software-defined, circuit switching network. We introduce a lower bound for a path flow in order to model a QoS demand for a single connection running the path. Instead of reducing the problem to the unsplittable flow problem, as suggested by Baier et al., we propose a potentially more exact method. We directly enhance the Raghavan and Thompson’s randomized rounding for ordinary MCF problems to account for k-splittability and the lower flow bounds. A mechanism is constructed that prevents rounding up low flows in the subproblem solution to big values. It is based on modifying the continuous subproblem by additionally penalizing flows of certain commodities in certain links. When k=1, this allows us to prove a property similar to the mathcal O(sqrt{log m}) approximation factor, where m denotes the number of network links. We give probabilistic guarantees for the solution quality and examine the behavior of the method experimentally.