Maximizing Stochastic Monotone Submodular Functions

TL;DR

This paper addresses maximizing stochastic monotone submodular functions under matroid constraints, introducing policies with provable approximation bounds and analyzing the adaptivity gap.

Contribution

It establishes the adaptivity gap as e/(e-1), proposes a polynomial-time non-adaptive policy achieving this bound, and analyzes the performance of adaptive and myopic policies.

Findings

The adaptivity gap is exactly e/(e-1).

A polynomial-time non-adaptive policy attains the adaptivity gap bound.

The myopic policy achieves a 1-1/e approximation ratio for uniform matroids.

Abstract

We study the problem of maximizing a stochastic monotone submodular function with respect to a matroid constraint. Due to the presence of diminishing marginal values in real-world problems, our model can capture the effect of stochasticity in a wide range of applications. We show that the adaptivity gap -- the ratio between the values of optimal adaptive and optimal non-adaptive policies -- is bounded and is equal to e/(e-1). We propose a polynomial-time non-adaptive policy that achieves this bound. We also present an adaptive myopic policy that obtains at least half of the optimal value. Furthermore, when the matroid is uniform, the myopic policy achieves the optimal approximation ratio of 1-1/e.