Online Max-min Fair Allocation

TL;DR

This paper investigates the online max-min fair allocation problem, providing asymptotically optimal algorithms for both adversarial and i.i.d. models, and introduces novel techniques for derandomization and nearly optimal solutions.

Contribution

It establishes the asymptotic competitive ratios for online max-min fair allocation and develops new algorithms with proven optimality and innovative derandomization methods.

Findings

Asymptotic $1/n$-competitive ratio for adversarial input, proven to be optimal.

A polynomial-time deterministic algorithm for i.i.d. inputs achieving near-optimal allocations.

Introduction of a novel derandomization technique that surpasses natural approaches.

Abstract

We study an online version of the max-min fair allocation problem for indivisible items. In this problem, items arrive one by one, and each item must be allocated irrevocably on arrival to one of $n$ agents, who have additive valuations for the items. Our goal is to make the least happy agent as happy as possible. In research on the topic of online allocation, this is a fundamental and natural problem. Our main result is to reveal the asymptotic competitive ratios of the problem for both the adversarial and i.i.d. input models. We design a polynomial-time deterministic algorithm that is asymptotically $1/n$-competitive for the adversarial model, and we show that this guarantee is optimal. To this end, we present a randomized algorithm with the same competitive ratio first and then derandomize it. A natural derandomization fails to achieve the competitive ratio of $1/n$. We instead build the algorithm by introducing a novel technique. When the items are drawn from an unknown identical and independent distribution, we construct a simple polynomial-time deterministic algorithm that outputs a nearly optimal allocation. We analyze the strict competitive ratio and show almost tight bounds for the solution. We further mention some implications of our results on variants of the problem.