Fixed-precision complexity of contiguous approximate envy-free cake cutting

Determine the computational complexity of finding a contiguous ε-envy-free allocation for general normalized rational piecewise-constant additive valuations when ε is any fixed value satisfying 0<ε≤1/4.

Background

The main theorem proves PPAD-completeness for contiguous ε-envy-free cake cutting under highly restricted additive valuations, but the hard instances use ε=Θ(n⁻⁴), where n is the number of agents. In contrast, prior work gives a polynomial-time algorithm for achieving an additive guarantee of 1/4+o(1). The paper therefore leaves unresolved the complexity at every fixed precision up to 1/4 for unrestricted normalized rational piecewise-constant additive valuations.