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.

References

The gap between Theorem 5.1 and Theorem 5.2 leaves open the following tantalizing question. Does there always exists an allocation of the cake with O(n) cuts that is envy-free w.r.t. the subjective utilities and the market valuation?

— Fair Division with Market Values  (2410.23137 - Barman et al., 2024) in Open Question, Section 5.1 (Cake Division – Number of Cuts)

In the cake-cutting setting, their result gives the upper bound $#1{n}\le n(n-1)$ for the stronger requirement of envy-freeness with respect to both the private valuations and the common market valuation, and their Open Question~5 asks whether $O(n)$ cuts always suffice. We do not settle that question.

— Exact Cut Complexity of Equal-Length Proportional Cake Cutting  (2610.19733 - Kawase et al., 14 Oct 2026) in Section 1, Related Work