Near-Optimal Online Resource Allocation in the Random-Order Model

We study the problem of allocating either divisible or indivisible items (goods or chores) among a set of agents, where the items arrive online, one at a time. Each agent's non-negative value for an item is set by an adversary upon the item's arrival. Our focus is on a unifying algorithmic framework for finding online allocations that treats both fairness and economic efficiency. For this sake, we aim to optimize the generalized means of agents' received values, covering a spectrum of welfare functions including average utilitarian welfare and egalitarian welfare. In the traditional adversarial model, where items arrive in an arbitrary order, no algorithm can give a decent approximation to welfare in the worst case. To escape from this strong lower bound, we consider the random-order model, where items arrive in a uniformly random order. This model provides us with a major breakthrough: we devise algorithms that guarantee a nearly-optimal competitive ratio for certain welfare functions, if the welfare obtained by the optimal allocation is sufficiently large. We prove that our results are almost tight: if the optimal solution's welfare is strictly below a certain threshold, then no nearly-optimal algorithm exists, even in the random-order model.