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

2025-08-25

AI Summary

This paper studies online set cover and load balancing under convex objectives, introducing the first algorithms designed directly for the **integral online setting**, bypassing the conventional paradigm of convex relaxation and dual analysis. Methodologically, it reformulates the problem as an online packing problem, decomposes it into decentralized multi-machine subproblems, and incorporates a randomized activation threshold mechanism to achieve fine-grained trade-offs between coverage and cost. Key contributions include: (1) the first extension of approximation guarantees under symmetric norms (e.g., ℓₚ, Top-k) from fractional to integral online settings; (2) generalization to generalized online scheduling with composite norm objectives; and (3) tight approximation ratios that improve upon or subsume prior results—most of which apply only to fractional solutions or offline settings.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Develops integral online algorithms for Set Cover
• Extends convex objectives to integral online settings
• Solves generalized scheduling with symmetric norms

Innovation

• Direct integral online algorithms bypassing primal-dual framework
• Reduction to online packing problems with approximation algorithms
• Random activation thresholds for efficient machine utilization