Towards optimal subsidy bounds for envy-freeable allocations

We study the fair division of indivisible items with subsidies among n agents, where the absolute marginal valuation of each item is at most one. Under monotone nondecreasing valuations (where each item is a good), Brustle et al. [9] demonstrated that a maximum subsidy of $2(n-1)$ and a total subsidy of $2{(n-1)}^2$ are sufficient to guarantee the existence of an envy-freeable allocation. In this paper, we improve upon these bounds, even in a wider model. Namely, we show that, given an EF1 allocation, we can compute in polynomial time an envy-free allocation with a subsidy of at most $n-1$ per agent and a total subsidy of at most $n(n-1)/2$. Moreover, when the valuations are monotone nondecreasing, we provide a polynomial-time algorithm that computes an envy-free allocation with a subsidy of at most $n-1.5$ per agent and a total subsidy of at most $(n^2-n-1)/2$.