Stochastic Makespan Minimization in Structured Set Systems
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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, 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, given a set of intervals in time, with each interval j having random load X_j, how do we choose t intervals to minimize the expected maximum load at any time? 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. Specifically, we give an O(loglog m)-approximation algorithm for all these problems. Our approach uses a strong LP relaxation using the cumulant generating functions of the random variables. We also show that this LP has an Ω(log^* m) integrality gap even for the problem of selecting intervals on a line. Moreover, we show logarithmic gaps for problems without geometric structure, showing that some structure is needed to get good results using these techniques.
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