Exact cut complexity of equal-length envy-freeness for arbitrary numbers of agents

Determine whether every instance of equal-length envy-free cake cutting with n agents admits an allocation using exactly 2n-2 cuts for arbitrary n, not only when n is a prime power.

Background

The paper proves that equal-length envy-free cake cutting requires at least 2n-2 cuts and achieves this bound when the number of agents is a prime power. For arbitrary n, it establishes only an O(n1.525) upper bound by repeatedly applying an exact-share splitting theorem and reducing to a nearby prime-power instance. Closing the gap between the lower bound 2n-2 and the general upper bound remains unresolved.

References

Two questions remain open: whether equal-length envy-freeness always admits $2n-2$ cuts, and whether a protocol in this augmented model attains $2n-2$ cuts for every $n$.

— Exact Cut Complexity of Equal-Length Proportional Cake Cutting  (2610.19733 - Kawase et al., 14 Oct 2026) in Section 6, Conclusion

Two questions remain open: whether equal-length envy-freeness always admits $2n-2$ cuts, and whether a protocol in this augmented model attains $2n-2$ cuts for every $n$.

— Exact Cut Complexity of Equal-Length Proportional Cake Cutting  (2610.19733 - Kawase et al., 14 Oct 2026) in Section 6, Conclusion