Concentration and maximin fair allocations for subadditive valuations
Uriel Feige, Shengyu Huang
TL;DR
The paper improves the MMS fair allocation guarantee for subadditive valuations from prior $\Theta(1/(\log n \log \log n))$ bounds to $\Theta(1/\log n)$ by refining the allocation procedure and leveraging new concentration bounds for subadditive valuations. The approach uses preprocessing to remove large items, Feige-style rounding across $t=\frac{56}{23}\log n$ copies, and uniform contention resolution to produce a valid allocation with high probability that each agent receives at least $1/(14\log n)$ of her MMS. Central to the analysis are concentration results of subadditive valuations (in particular, a bound on $\mathbb{E}[v(S')]$ in terms of the median $M$ and the maximum item value $b$) which enable the probabilistic guarantees needed for the contention-resolution step. The findings also establish near-tightness of the upper bound in certain regimes and discuss potential avenues for further improvements, including algorithmic changes beyond the current rounding framework.
Abstract
We consider fair allocation of $m$ indivisible items to $n$ agents of equal entitlements, with submodular valuation functions. Previously, Seddighin and Seddighin [{\em Artificial Intelligence} 2024] proved the existence of allocations that offer each agent at least a $\frac{1}{c \log n \log\log n}$ fraction of her maximin share (MMS), where $c$ is some large constant (over 1000, in their work). We modify their algorithm and improve its analysis, improving the ratio to $\frac{1}{14 \log n}$. Some of our improvement stems from tighter analysis of concentration properties for the value of any subadditive valuation function $v$, when considering a set $S' \subseteq S$ of items, where each item of $S$ is included in $S'$ independently at random (with possibly different probabilities). In particular, we prove that up to less than the value of one item, the median value of $v(S')$, denoted by $M$, is at least two-thirds of the expected value, $M \geq \frac{2}{3}\E[v(S')] - \frac{11}{12}\max_{e \in S} v(e)$.