Dependence on the number of agents and approximation precision

Determine the dependence on the number of agents n in the complexity of computing contiguous ε-envy-free cake allocations between the PPAD-hardness regime ε=Θ(n⁻⁴) and the polynomial-time guarantee ε=M/n obtained by equal division under density bound M.

Background

The paper presents PPAD-hardness for instances with approximation parameter ε=Θ(n⁻⁴), while showing that equal division yields an ε-envy-free allocation when ε is at least M/n under a density bound ρᵢ≤M. These results leave a gap in the dependence of computational complexity on n and ε. The authors explicitly identify the characterization of this intermediate regime as unresolved.