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.