Integral Online Algorithms for Set Cover and Load Balancing with Convex Objectives

Online Set Cover and Load Balancing are central problems in online optimization, and there is a long line of work on developing algorithms for these problems with convex objectives. Although we know optimal online algorithms with $\ell_p$-norm objectives, recent developments for general norms and convex objectives that rely on the online primal-dual framework apply only to fractional settings due to large integrality gaps. Our work focuses on directly designing integral online algorithms for Set Cover and Load Balancing with convex objectives, bypassing the convex-relaxation and the primal-dual technique. Some of the main implications are: 1. For Online Set Cover, we can extend the results of Azar et. al. (2016) for convex objectives and of Kesselheim, Molinaro, and Singla (2024) for symmetric norms from fractional to integral settings. 2. Our results for convex objectives and symmetric norms even apply to the online generalized scheduling problem, which generalizes both Set Cover and Load Balancing. Previous works could only handle the offline version of this problem with norm objectives (Deng, Li, and Rabani 2023). 3. Our methods easily extend to settings with disjoint-composition of norms. This allows us to recover or improve the norm-composition results of Nagarajan and Shen (2020), and Kesselheim, Molinaro, and Singla (2024), and to extend our results to a large class of norms beyond symmetric. Our approach is to first reduce these problems to online packing problems, and then to design good approximation algorithms for the latter. To solve these packing problems, we use two key ideas. First, we decouple the global packing problem into a series of local packing problems on different machines. Next, we choose random activation thresholds for machines such that conditional on a machine being activated, the expected number of jobs it covers is high compared to its cost.