Stochastic makespan minimization in structured set systems

We study stochastic combinatorial optimization problems where the objective is to minimize expected maximum load (a.k.a. makespan). In this framework, we have a set of n tasks and m resources, each task j uses some subset resources. Tasks random sizes $$X_j$$ , our goal non-adaptively select t over all on any resource i total size selected that use i. For example, when resources are points intervals in line, obtain an $$O(\log \log m)$$ -approximation algorithm. Our technique also applicable other with geometric structure relation between resources; e.g., packing paths, rectangles, “fat” objects. approach strong LP relaxation using cumulant generating functions variables. show has $$\varOmega (\log ^* integrality gap, even for problem selecting line; here $$\log m$$ iterated logarithm function.