Approximation Algorithms for Stochastic Minimum Norm Combinatorial Optimization

Motivated by the need for, and growing interest in, modeling uncertainty in data, we introduce and study {\em stochastic minimum-norm optimization}. We have an underlying combinatorial optimization problem where the costs involved are {\em random variables} with given distributions; each feasible solution induces a random multidimensional cost vector, and given a certain objective function, the goal is to find a solution (that does not depend on the realizations of the costs) that minimizes the expected objective value. For instance, in stochastic load balancing, jobs with random processing times need to be assigned to machines, and the induced cost vector is the machine-load vector. Recently, in the deterministic setting, Chakrabarty and Swamy \cite{ChakrabartyS19a} considered a fairly broad suite of objectives, wherein we seek to minimize the $f$-norm of the cost vector under a given {\em arbitrary monotone, symmetric norm} $f$. In stochastic minimum-norm optimization, we work with this broad class of objectives, and seek a solution that minimizes the {\em expected $f$-norm} of the induced cost vector. We give a general framework for devising algorithms for stochastic minimum-norm combinatorial optimization, using which we obtain approximation algorithms for the stochastic minimum-norm versions of the load balancing and spanning tree problems. Two key technical contributions of this work are: (1) a structural result of independent interest connecting stochastic minimum-norm optimization to the simultaneous optimization of a (\emph{small}) collection of expected $\mathsf{Top}$-$\ell$-norms; and (2) showing how to tackle expected $\mathsf{Top}$-$\ell$-norm minimization by leveraging techniques used to deal with minimizing the expected maximum, circumventing the difficulties posed by the non-separable nature of $\mathsf{Top}$-$\ell$ norms.