Generalized Load Balancing and Clustering Problems with Norm Minimization

In many fundamental combinatorial optimization problems, a feasible solution induces some real cost vectors as an intermediate result, and the optimization objective is a certain function of the vectors. For example, in the problem of makespan minimization on unrelated parallel machines, a feasible job assignment induces a vector containing the sizes of assigned jobs for each machine, and the goal is to minimize the $L_\infty$ norm of $L_1$ norms of the vectors. Another example is fault-tolerant $k$-center, where each client is connected to multiple open facilities, thus having a vector of distances to these facilities, and the goal is to minimize the $L_\infty$ norm of $L_\infty$ norms of these vectors. In this paper, we study the maximum of norm problem. Given an arbitrary symmetric monotone norm $f$, the objective is defined as the maximum ($L_\infty$ norm) of $f$-norm values of the induced cost vectors. This versatile formulation captures a wide variety of problems, including makespan minimization, fault-tolerant $k$-center and many others. We give concrete results for load balancing on unrelated parallel machines and clustering problems, including constant-factor approximation algorithms when $f$ belongs with a certain rich family of norms, and $O(\log n)$-approximations when $f$ is general and satisfies some mild assumptions. We also consider the aforementioned problems in a generalized fairness setting. As a concrete example, the insight is to prevent a scheduling algorithm from assigning too many jobs consistently on any machine in a job-recurring scenario, and causing the machine's controller to fail. Our algorithm needs to stochastically output a feasible solution minimizing the objective function, and satisfy the given marginal fairness constraints.