Stochastic makespan minimization in structured set systems

Anupam Gupta, Amit Kumar, Viswanath Nagarajan, Xiangkun Shen

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