A Tight Bound for Stochastic Submodular Cover

TL;DR

This paper proves a tight approximation bound for the Adaptive Greedy algorithm in Stochastic Submodular Cover, improving previous bounds and generalizing classical Set Cover results.

Contribution

It establishes a new tight bound of $( ext{ln}(Q/ ext{eta})+1)$ for the adaptive greedy algorithm, correcting earlier proofs and extending classical set cover bounds.

Findings

The bound $( ext{ln}(Q/ ext{eta})+1)$ is tight for the problem.

Previous bounds were quadratic or larger, now improved to a tight bound.

The result generalizes the classical set cover approximation bound.

Abstract

We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) achieves an approximation bound of $(\ln (Q/\eta)+1)$ for Stochastic Submodular Cover: here $Q$ is the "goal value" and $\eta$ is the smallest non-zero marginal increase in utility deliverable by an item. (For integer-valued utility functions, we show a bound of $H(Q)$, where $H(Q)$ is the $Q^{th}$ Harmonic number.) Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan and Saligrama (2017). The subsequent corrected proof of Golovin and Krause (2017) gives a quadratic bound of $(\ln(Q/\eta) + 1)^2$. Other previous bounds for the problem are $56(\ln(Q/\eta) + 1)$, implied by work of Im et al. (2016) on a related problem, and $k(\ln (Q/\eta)+1)$, due to Deshpande et al. (2016) and Hellerstein and Kletenik (2018), where $k$ is the number of states. Our bound generalizes the well-known $(\ln~m + 1)$ approximation bound on the greedy algorithm for the classical Set Cover problem, where $m$ is the size of the ground set.