Differentially Private Multicolor Discrepancy and Fair Division of Indivisible Goods

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We study the fair division of indivisible goods under pure differential privacy, continuing the line of work initiated by Manurangsi and Suksompong. For $n$ agents with nonnegative additive utilities over $m$ goods and a fixed privacy parameter, we give an entry-private algorithm that, with high probability, achieves consensus envy-freeness up to $O(\sqrt n+\log^3 m)$ goods. This substantially improves the dependence on $n$ over the previous $O(n\log m)$ guarantee for ordinary envy-freeness, while providing the stronger consensus guarantee. A key ingredient is a private algorithm for multicolor discrepancy, which may be of independent interest. Our algorithm may require exponential time. We also obtain substantially stronger guarantees under additional structure: when all item values belong to a public alphabet of size $D$, we give a polynomial-time entry-private algorithm achieving ordinary envy-freeness up to $O(\operatorname{polylog}(mD))$ goods with high probability. Finally, we prove an $Ω(\log n)$ lower bound on the number of goods that must be removed to achieve ordinary envy-freeness under entry privacy, for sufficiently many goods, even with binary utilities.