Minimum Cost Adaptive Submodular Cover

TL;DR

This paper introduces a near-optimal approximation algorithm for the minimum cost adaptive submodular cover problem, extending to multiple functions and various cost moments, with proven theoretical guarantees and empirical validation.

Contribution

It presents a novel approximation algorithm for adaptive submodular cover that achieves near-optimal guarantees and extends to multiple functions and different cost moments.

Findings

The algorithm achieves a $4(1+ ext{ln} Q)$ approximation ratio.

The method extends to minimizing the $p^{th}$ moment of coverage cost.

Empirical results demonstrate effectiveness on hypothesis identification instances.

Abstract

Adaptive submodularity is a fundamental concept in stochastic optimization, with numerous applications such as sensor placement, hypothesis identification and viral marketing. We consider the problem of minimum cost cover of adaptive-submodular functions, and provide a $4(1+\ln Q)$-approximation algorithm, where $Q$ is the goal value. In fact, we consider a significantly more general objective of minimizing the $p^{th}$ moment of the coverage cost, and show that our algorithm simultaneously achieves a $(p+1)^{p+1}\cdot (\ln Q+1)^p$ approximation guarantee for all $p\ge 1$. All our approximation ratios are best possible up to constant factors (assuming $P\ne NP$). Moreover, our results also extend to the setting where one wants to cover {\em multiple} adaptive-submodular functions. Finally, we evaluate the empirical performance of our algorithm on instances of hypothesis identification.