Online Allocation with Concave, Diminishing-Returns Objectives

2025-10-13

AI Summary

This paper studies online resource allocation under concave objective functions exhibiting diminishing marginal returns: items arrive sequentially and must be allocated fractionally and irrevocably to multiple agents to maximize total reward. We propose a general algorithm grounded in the online primal-dual framework, which performs continuous greedy allocation by constructing a “balanced” auxiliary objective function (U(mathbf{x})). We establish, for the first time, that for *all* online resource allocation problems with such concavity properties, there exists a fractional algorithm achieving a competitive ratio of (1 - 1/e approx 0.632)—matching the fundamental information-theoretic lower bound. Our work unifies and generalizes prior isolated results tailored to specific concave functions (e.g., square root, logarithm), thereby providing the first universal, optimal fractional online algorithmic framework for this broad class of problems.

Technology Category

Machine Learning: Online Learning & BanditsSearch and Optimization: Algorithm ConfigurationMultiagent Systems: Mechanism Design

Application Category

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

Abstract

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.

Problem

• Designing competitive algorithms for online resource allocation problems
• Maximizing concave diminishing-returns objectives with item arrivals
• Achieving optimal competitive ratios for fractional allocation settings

Innovation

• Continuous greedy allocations using auxiliary objective function
• Online primal-dual method with balanced property
• General 1-1/e-competitive algorithm for concave objectives