Approximate Proportionality in Online Fair Division 文章

arXiv:2508.03253v2 Announce Type: replace-cross Abstract: We study the online fair division problem, where indivisible goods arrive sequentially and must be allocated immediately and irrevocably. Prior work establishes strong impossibility results for approximating classic notions such as envy-freeness up to one good (EF1) and maximin share (MMS) in this setting, but the approximability of proportionality up to one good (PROP1) has remained unresolved. We resolve this gap in two steps. First, we show that three natural greedy allocation rules (standard baselines in fair division) fail to guarantee any multiplicative approximation to PROP1 against an adaptive adversary. These limitations motivate two relaxations: (i) restricting attention to a non-adaptive adversary, and (ii) incorporating coarse predictions in the spirit of learning-augmented algorithms.