Cake division with O(n) cuts achieving EF with respect to agents and market values

Determine whether, for every cake-cutting instance with n agents’ subjective utilities and a market valuation, there exists a division using O(n) cuts that is envy-free (EF) with respect to the subjective utilities and simultaneously EF with respect to the market valuation.

Background

Using Alon’s necklace-splitting theorem, the paper proves existence of cake divisions that are EF for agents and market values with at most n(n−1) cuts, and provides a 2n−2 lower bound even for identical utilities. This leaves a quadratic-vs.-linear gap in the number of cuts.

The open question asks whether the stronger guarantee of EF on both sides can be achieved with only O(n) cuts, exploiting the additional flexibility of needing EF (not perfect equal division) for subjective utilities.