Stochastic unsplittable flows

We consider the stochastic unsplittable flow problem: given a graph with edge-capacities, k source-sink pairs with each pair {sj , tj} having a size Sj and value vj , the goal is to route the pairs unsplittably while respecting edge capacities to maximize the total value of the routed pairs. However, the size Sj is a random variable and is revealed only after we decide to route pair j. Which pairs should we route, along which paths, and in what order so as to maximize the expected value? We present results for several cases of the problem under the no-bottleneck assumption. We show a logarithmic approximation algorithm for the single-sink problem on general graphs, considerably improving on the prior results of Chawla and Roughgarden which worked for planar graphs. We present an approximation to the stochastic unsplittable flow problem on directed acyclic graphs, within less than a logarithmic factor of the best known approximation in the nonstochastic setting. We present a non-adaptive strategy on trees that is within a constant factor of the best adaptive strategy, asymptotically matching the best results for the non-stochastic unsplittable flow problem on trees. Finally, we give results for the stochastic unsplittable flow problem on general graphs. Our techniques include using edge-confluent flows for the single-sink problem in order to control the interaction between flow-paths, and a reduction from general scheduling policies to "safe" ones (i.e., those guaranteeing no capacity violations), which may be of broader interest.