Efficient optimization for modified fractional expectation thresholds

Develop an efficient algorithm to compute an optimizer for the modified fractional expectation threshold ψ_q for arbitrary monotone properties, and determine whether an onion-peeling process analogous to the graph onion decomposition exists for every monotone property.

Background

For arbitrary monotone properties planted in Bernoulli noise, the paper introduces modified fractional expectation thresholds ψ_q and proves that these thresholds determine the minimax estimation rate up to asymptotic multiplicative separation.

In contrast to the weakly dense graph case, where the onion decomposition yields polynomial-time computation of the corresponding thresholds, the authors do not provide an efficient method for arbitrary monotone properties. They explicitly identify both efficient optimization and the existence of a general onion-peeling analogue as unresolved questions.

References

We remark that while we are able to efficiently compute $\varphi_q$ in the setting of weakly dense graphs using the onion decomposition of $H$, we do not yet have a way to efficiently compute an optimizer for $\psi_q$ for arbitrary monotone properties. It is an interesting question for future work if a similar onion-peeling process exists for any monotone property.

— The Fundamental Limits of Recovering Planted Subgraphs  (2503.15723 - Lee et al., 19 Mar 2025) in Section 5, “Characterizing the Minimax curve via the ψ_q thresholds,” concluding remark