Stochastic $\ell_p$ Load Balancing and Moment Problems via the $L$-Function Method

This paper considers stochastic optimization problems whose objective functions involve powers of random variables. For example, consider the classic Stochastic lp Load Balancing Problem (SLBp): There are $m$ machines and $n$ jobs, and known independent random variables $Y_{ij}$ decribe the load incurred on machine $i$ if we assign job $j$ to it. The goal is to assign each jobs to machines in order to minimize the expected $l_p$-norm of the total load on the machines. While convex relaxations represent one of the most powerful algorithmic tools, in problems such as SLBp the main difficulty is to capture the objective function in a way that only depends on each random variable separately. We show how to capture $p$-power-type objectives in such separable way by using the $L$-function method, introduced by Latała to relate the moment of sums of random variables to the individual marginals. We show how this quickly leads to a constant-factor approximation for very general subset selection problem with $p$-moment objective. Moreover, we give a constant-factor approximation for SLBp, improving on the recent $O(p/\ln p)$-approximation of [Gupta et al., SODA 18]. Here the application of the method is much more involved. In particular, we need to sharply connect the expected $l_p$-norm of a random vector with the $p$-moments of its marginals (machine loads), taking into account simultaneously the different scales of the loads that are incurred by an unknown assignment.