In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In k-clustering, opening k facilities induces an assignment cost vector across the clients. Typically, one seeks a solution which either minimizes the sum- or the max- of this vector, and these problems (makespan minimization, k-median, and k-center) are classic NP-hard problems which have been extensively studied.

In this paper we consider the minimum-norm optimization problem. Given an arbitrary monotone, symmetric norm, the problem asks to find a solution which minimizes the norm of the induced cost-vector. Such norms are versatile and include ℓ_p norms, Top-ℓ norm (sum of the ℓ largest coordinates in absolute value), and ordered norms (non-negative linear combination of Top-ℓ norms), and consequently, the minimum-norm problem models a wide variety of problems under one umbrella, We give a general framework to tackle the minimum-norm problem, and illustrate its efficacy in the unrelated machine load balancing and k-clustering setting. Our concrete results are the following.

(a) We give constant factor approximation algorithms for the minimum norm load balancing problem in unrelated machines, and the minimum norm k-clustering problem. To our knowledge, our results constitute the first constant-factor approximations for such a general suite of objectives.

(b) For load balancing on unrelated machines, we give a (2+ε)-approximation for ordered load balancing (i.e., min-norm load-balancing under an ordered norm).

(c) For k-clustering, we give a (5+ε)-approximation for the ordered k-median problem, which significantly improves upon the previous-best constant-factor approximation (Chakrabarty and Swamy (ICALP 2018); Byrka, Sornat, and Spoerhase (STOC 2018)).

(d) Our techniques also imply O(1) approximations to the instance-wise best simultaneous approximation factor for unrelated-machine load-balancing and k-clustering. To our knowledge, these are the first positive simultaneous approximation results in these settings.

At a technical level, one of our chief insights is that minimum-norm optimization can be reduced to a special case that we call min-max ordered optimization. Both the reduction, and the task of devising algorithms for the latter problem, require a sparsification idea that we develop, which is of interest for ordered optimization problems. The main ingredient in solving min-max ordered optimization is a deterministic, oblivious rounding procedure (that we devise) for suitable LP relaxations of the load-balancing and k-clustering problem; this may be of independent interest.