Tighten bounds for EFd and PROPd in group fair division

Determine tight asymptotic bounds on the minimum d guaranteeing the existence of envy-freeness up to d items, EF(n1,...,nk), and proportionality up to d items, PROP(n1,...,nk), in the k-group setting by closing the gaps between the best known upper bounds and the new lower bounds established here, namely Ω(√(n/(k ln k)) for EF and Ω(√(n/(k^3 ln k))) as well as Ω(√(min{n1,...,nk}/ln k)) for PROP.

Background

For group fair division with n agents partitioned into k groups, the paper establishes new lower bounds: EFd may be infeasible for d in Ω(√(n/(k ln k))), while PROPd may be infeasible for d in Ω(√(n/(k^3 ln k))) and also Ω(√(min{n1,...,nk}/ln k)).

Known positive results (e.g., EFd allocations exist for d in O(√n)) imply larger remaining gaps than in the discrepancy/CD case. The authors explicitly identify closing these gaps as open problems.