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 stochastic minimum-norm optimization. We have an underlying combinatorial optimization problem where the costs involved are 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. The choice of objective is typically the maximum-or sum-of the entries of the cost vector, or in some cases some other ℓ p norm of the cost vector. Recently, in the deterministic setting, Chakrabarty and Swamy [8] considered a much broader suite of objectives, wherein we seek to minimize the f -norm of the cost vector under a given 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 expected f -norm of the induced cost vector.

The class of monotone, symmetric norms is versatile and includes ℓ p -norms, and Top ℓ -norms (sum of ℓ largest coordinates in absolute value), and enjoys various closure properties; in particular, it can be used to incorporate multiple norm budget constraints f ℓ (x) ≤ B ℓ , ℓ = 1, . . . , k.

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. We obtain the following concrete results.

• An O(1)-approximation for stochastic minimum-norm load balancing on unrelated machines with:

(i) arbitrary monotone symmetric norms and job sizes that are Bernoulli random variables; and (ii) Top ℓ norms and arbitrary job-size distributions.

• An O (log m/ log log m)-approximation for the general stochastic minimum-norm load balancing problem, where m is the number of machines.

• An O(1)-approximation for stochastic minimum-norm spanning tree with arbitrary monotone symmetric norms and distributions; this guarantee extends to the stochastic minimum-norm matroid basis problem.

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 (small) collection of expected Top ℓ -norms; and (2) showing how to tackle expected Top ℓ -norm minimization by leveraging techniques used to deal with minimizing the expected maximum, circumventing the difficulties posed by the non-separable nature of Top ℓ norms.