Online Allocation with Concave, Diminishing-Returns Objectives

Online resource allocation problems are central challenges in economics and computer science, modeling situations in which $n$ items arriving one at a time must each be immediately allocated among $m$ agents. In such problems, our objective is to maximize a monotone reward function $f(\mathbf{x})$ over the allocation vector $\mathbf{x} = (x_{ij})_{i, j}$, which describes the amount of each item given to each agent. In settings where $f$ is concave and has "diminishing returns" (monotone decreasing gradient), several lines of work over the past two decades have had great success designing constant-competitive algorithms, including the foundational work of Mehta et al. (2005) on the Adwords problem and many follow-ups. Notably, while a greedy algorithm is $\frac{1}{2}$-competitive in such settings, these works have shown that one can often obtain a competitive ratio of $1-\frac{1}{e} \approx 0.632$ in a variety of settings when items are divisible (i.e. allowing fractional allocations). However, prior works have thus far used a variety of problem-specific techniques, leaving open the general question: Does a $(1-\frac{1}{e})$-competitive fractional algorithm always exist for online resource allocation problems with concave, diminishing-returns objectives? In this work, we answer this question affirmatively, thereby unifying and generalizing prior results for special cases. Our algorithm is one which makes continuous greedy allocations with respect to an auxiliary objective $U(\mathbf{x})$. Using the online primal-dual method, we show that if $U$ satisfies a "balanced" property with respect to $f$, then one can bound the competitiveness of such an algorithm. Our crucial observation is that there is a simple expression for $U$ which has this balanced property for any $f$, yielding our general $(1-\frac{1}{e})$-competitive algorithm.