Online Algorithms for the Santa Claus Problem

TL;DR

This paper investigates online algorithms for the Santa Claus problem, demonstrating that near-optimal solutions are achievable under random item arrival, with tight bounds on performance depending on the optimal assignment value.

Contribution

The paper introduces a competitive ratio result for online Santa Claus problem under random arrivals and establishes near-tight bounds for algorithm performance.

Findings

Achieves a (1-ε)-competitive ratio for large enough optimal values under random arrivals.

Shows no (1-ε)-competitive algorithm exists for smaller optimal values, establishing tight bounds.

Provides theoretical analysis of online fair division with probabilistic item arrival models.

Abstract

The Santa Claus problem is a fundamental problem in fair division: the goal is to partition a set of heterogeneous items among heterogeneous agents so as to maximize the minimum value of items received by any agent. In this paper, we study the online version of this problem where the items are not known in advance and have to be assigned to agents as they arrive over time. If the arrival order of items is arbitrary, then no good assignment rule exists in the worst case. However, we show that, if the arrival order is random, then for $n$ agents and any $\varepsilon > 0$, we can obtain a competitive ratio of $1-\varepsilon$ when the optimal assignment gives value at least $\Omega(\log n / \varepsilon^2)$ to every agent (assuming each item has at most unit value). We also show that this result is almost tight: namely, if the optimal solution has value at most $C \ln n / \varepsilon$ for some constant $C$, then there is no $(1-\varepsilon)$-competitive algorithm even for random arrival order.