Cost of consistency for polynomial-time randomized algorithms

Determine whether imposing per-step consistency—restricting an online algorithm for monotone submodular maximization under a cardinality constraint to make at most a constant number of changes per insertion—incurs a fundamental loss in the best achievable approximation ratio by polynomial-time randomized algorithms. Concretely, ascertain whether the optimal approximation ratio for polynomial-time randomized consistent algorithms is strictly below the classical offline (1−1/e) barrier or can match it.

Background

The paper presents a consistent polynomial-time randomized algorithm achieving a 0.51-approximation, while noting that, assuming P ≠ NP, no polynomial-time algorithm (consistent or not) can surpass the (1−1/e) ≈ 0.64 threshold in the offline setting. The information-theoretic bounds established in the paper concern non-polynomial-time algorithms.

The authors explicitly pose whether a "cost of consistency" persists for polynomial-time randomized algorithms—that is, whether the consistency constraint strictly lowers the achievable approximation factor relative to the best possible in the offline setting.

References

We leave as an intriguing open problem, whether there is a "cost of consistency" for poly-time randomized algorithms.

— The Cost of Consistency: Submodular Maximization with Constant Recourse  (2412.02492 - Dütting et al., 2024) in Introduction, Separation for Efficient Algorithms

Several quantitative questions remain open. What is the minimum hard recourse needed to attain $-\varepsilon$ as $\varepsilon\downarrow0$---in particular, must it diverge?

— A Sharp Barrier for Consistent Submodular Maximization: Any Improvement over $2-\sqrt{2}$ Entails Exponential Queries or Linear Recourse  (2609.09986 - Fu et al., 9 Sep 2026) in Section 6, Discussion