Equitable allocation of indivisible goods to agents in online settings is an algorithmic primitive with applications for load balancing, network routing, online marketplaces, and multi-agent systems. We consider a general setting in which allocations are constrained to be bases of discrete polymatroids that arrive online. Our work demonstrates that a simple, myopic algorithm called Brick-Laying, which greedily minimizes the sum of squared loads on agents, achieves a universal and objective-free notion of optimality called majorization minimax-optimality [BDK26] for this setting. As a consequence, Brick-Laying simultaneously guarantees minimax optimal competitive ratios and regret for all Schur-concave and Schur-convex objectives, and for any number of agents and resources (despite being agnostic to problem scale). Departing from popular primal-dual analysis, we employ majorization to compare allocations. We leverage the conjugates of integer partitions -- which act as a discrete dual to majorization -- to characterize worst-case instances for the Brick-Laying algorithm. Our approach reveals a novel structural connection between the geometry of partitions and online equitable allocation.

翻译：在线环境下向智能体公平分配不可分物品是算法设计的基本问题，广泛应用于负载均衡、网络路由、在线市场和多智能体系统。本文考虑一种一般性设置：分配受限于在线到达的离散多拟阵基。研究表明，名为"砌砖法"的简单贪婪算法通过最小化智能体负载平方和，在此设置下实现了名为"主序极小化最优性"的通用无目标最优性概念。因此，该算法对所有Schur-凹和Schur-凸目标函数、任意数量的智能体与资源（尽管对问题规模无感知）均能同时保证极小化最优竞争比与后悔值。与流行的原始-对偶分析不同，我们采用主序方法比较分配方案，并利用整数划分的共轭（作为主序的离散对偶）来刻画"砌砖法"算法的最坏情形实例。本文揭示了划分几何结构与在线公平分配之间新颖的结构性关联。