Adaptive Submodular Optimization under Matroid Constraints

TL;DR

This paper extends the classic greedy algorithm for submodular maximization under matroid constraints to adaptive settings with partial observability, providing approximation guarantees.

Contribution

It introduces an adaptive greedy algorithm with provable approximation bounds for maximizing adaptive monotone submodular functions under multiple matroid constraints.

Findings

The adaptive greedy algorithm achieves a 1/(p+1) approximation ratio.

The results apply to arbitrary p-independence systems.

Application demonstrated on adaptive match-making problem.

Abstract

Many important problems in discrete optimization require maximization of a monotonic submodular function subject to matroid constraints. For these problems, a simple greedy algorithm is guaranteed to obtain near-optimal solutions. In this article, we extend this classic result to a general class of adaptive optimization problems under partial observability, where each choice can depend on observations resulting from past choices. Specifically, we prove that a natural adaptive greedy algorithm provides a $1/(p+1)$ approximation for the problem of maximizing an adaptive monotone submodular function subject to $p$ matroid constraints, and more generally over arbitrary $p$-independence systems. We illustrate the usefulness of our result on a complex adaptive match-making application.