Lower Bound on the Greedy Approximation Ratio for Adaptive Submodular Cover

TL;DR

This paper establishes a lower bound of approximately 1.3*(1+ln Q) on the approximation ratio of the greedy algorithm for adaptive submodular cover, challenging previous claims of a (1+ln Q)^2 bound.

Contribution

It provides the first known lower bound for the greedy algorithm's approximation ratio in adaptive submodular cover, invalidating prior upper bound claims.

Findings

Greedy algorithm's approximation ratio is at least 1.3*(1+ln Q).

The constructed instance with Q=1 shows the bound's tightness.

Previous (1+ln Q)^2 approximation ratio claim is invalidated.

Abstract

We show that the greedy algorithm for adaptive-submodular cover has approximation ratio at least 1.3*(1+ln Q). Moreover, the instance demonstrating this gap has Q=1. So, it invalidates a prior result in the paper ``Adaptive Submodularity: A New Approach to Active Learning and Stochastic Optimization'' by Golovin-Krause, that claimed a (1+ln Q)^2 approximation ratio for the same algorithm.