Parameterized Guarantees for Almost Envy-Free Allocations

We study fair allocation of indivisible goods among agents with additive valuations. We obtain novel approximation guarantees for three of the strongest fairness notions in discrete fair division, namely envy-free up to the removal of any positively-valued good (EFx), pairwise maximin shares (PMMS), and envy-free up to the transfer of any positively-valued good (tEFx). Our approximation guarantees are in terms of an instance-dependent parameter $\gamma \in (0,1]$ that upper bounds, for each indivisible good in the given instance, the multiplicative range of nonzero values for the good across the agents. First, we consider allocations wherein, between any pair of agents and up to the removal of any positively-valued good, the envy is multiplicatively bounded. Specifically, the current work develops a polynomial-time algorithm that computes a $\left( \frac{2\gamma}{\sqrt{5+4\gamma}-1}\right)$-approximately EFx allocation for any given fair division instance with range parameter $\gamma \in (0,1]$. For instances with $\gamma \geq 0.511$, the obtained approximation guarantee for EFx surpasses the previously best-known approximation bound of $(\phi-1) \approx 0.618$, here $\phi$ denotes the golden ratio. Furthermore, for $\gamma \in (0,1]$, we develop a polynomial-time algorithm for finding allocations wherein the PMMS requirement is satisfied, between every pair of agents, within a multiplicative factor of $\frac{5}{6} \gamma$. En route to this result, we obtain novel existential and computational guarantees for $\frac{5}{6}$-approximately PMMS allocations under restricted additive valuations. Finally, we develop an algorithm that efficiently computes a $2\gamma$-approximately tEFx allocation. Specifically, we obtain existence and efficient computation of exact tEFx allocations for all instances with $\gamma \in [0.5, 1]$.