Discrete Stochastic Submodular Maximization: Adaptive vs. Non-Adaptive vs. Offline

TL;DR

This paper investigates the differences in performance between adaptive, non-adaptive, and offline algorithms for stochastic monotone submodular maximization, providing bounds on the adaptivity gap and approximation factors under various constraints.

Contribution

It introduces a transformation from adaptive decision trees to non-adaptive chains, establishing bounds on the adaptivity gap and analyzing approximation ratios for specific constraints.

Findings

The adaptivity gap is at most 1/τ, which is 2 for uniform distributions.

A simple adaptive greedy algorithm achieves a (1 - 1/e^τ) approximation for cardinality constraints.

Non-adaptive solutions can approximate the optimal adaptive and offline solutions within specific factors.

Abstract

We consider the problem of stochastic monotone submodular function maximization, subject to constraints. We give results on adaptivity gaps, and on the gap between the optimal offline and online solutions. We present a procedure that transforms a decision tree (adaptive algorithm) into a non-adaptive chain. We prove that this chain achieves at least ${\tau}$ times the utility of the decision tree, over a product distribution and binary state space, where ${\tau} = \min_{i,j} \Pr[x_i=j]$. This proves an adaptivity gap of $1/{\tau}$ (which is $2$ in the case of a uniform distribution) for the problem of stochastic monotone submodular maximization subject to state-independent constraints. For a cardinality constraint, we prove that a simple adaptive greedy algorithm achieves an approximation factor of $(1-1/e^{\tau})$ with respect to the optimal offline solution; previously, it has been proven that the algorithm achieves an approximation factor of $(1-1/e)$ with respect to the optimal adaptive online solution. Finally, we show that there exists a non-adaptive solution for the stochastic max coverage problem that is within a factor $(1-1/e)$ of the optimal adaptive solution and within a factor of ${\tau}(1-1/e)$ of the optimal offline solution.