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 $X_j$, 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 $O(\log\log m)$-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 $\Omega(\log^* m)$ integrality gap, even for the problem of selecting intervals on a line; here $\log^* m$ is the iterated logarithm function.