Almost and Approximate EFX for Few Types of Agents

We study the problem of fair allocation of a set of indivisible goods among $n$ agents with $k$ distinct additive valuations, with the goal of achieving approximate envy-freeness up to any good ($\alpha-\mathrm{EFX}$). It is known that EFX allocations exist for $n$ agents when there are at most three distinct valuations due to HV et al. Furthermore, Amanatidis et al. showed that a $\frac{2}{3}-\mathrm{EFX}$ allocation is guaranteed to exist when number of agents is at most seven. In this paper, we show that a $\frac{2}{3}-\mathrm{EFX}$ allocation exists for any number of agents when there are at most four distinct valuations. Secondly, we consider a relaxation called $\mathrm{EFX}$ with charity, where some goods remain unallocated such that no agent envies the set of unallocated goods. Akrami et al. showed that for $n$ agents and any $\varepsilon \in \left(0, \frac{1}{2}\right]$, there exists a $(1-\varepsilon)-\mathrm{EFX}$ allocation with at most $\tilde{\mathcal{O}}((n/\varepsilon)^{\frac{1}{2}})$ goods to charity. In this paper, we show that a $(1-\varepsilon)-\mathrm{EFX}$ allocation with a $\tilde{\mathcal{O}}(k/\varepsilon)^{\frac{1}{2}}$ charity exists for any number of agents when there are at most $k$ distinct valuations.