Subexponential or polynomial envy-free cake-cutting protocol

Determine whether improving the query bounds for the Aziz–Mackenzie subroutines or avoiding calls to those subroutines involving a linear number of agents can yield a subexponential or polynomial Robertson–Webb protocol for complete envy-free cake cutting with arbitrary nonatomic additive valuations.

Background

The paper establishes an upper bound of n{O(1)}2n Robertson–Webb queries for computing a complete envy-free allocation. Its analysis attributes the exponential factor primarily to calls to the Aziz–Mackenzie subroutines involving Θ(n) agents. The authors identify two possible routes toward improvement: reducing the query complexity of those subroutines or redesigning the protocol so that each call involves o(n) agents. Either development would potentially reduce the overall complexity to subexponential, or even polynomial, in the number of agents.

References

Naturally, this suggests two ways forward. One is to improve the query bounds for these subroutines, and the other is to avoid calling them with so many agents. Indeed, if every call involved $o(n)$ agents, then our protocol would use $2{o(n)}$ queries. Can either approach yield a subexponential or even polynomial protocol?

— Cutting Down the Tower: Single-Exponential Envy-Free Cake Cutting  (2609.05191 - Ye et al., 4 Sep 2026) in Section 5, Discussion and Concluding Remarks