Non-Submodular Maximization via the Greedy Algorithm and the Effects of Limited Information in Multi-Agent Execution

TL;DR

This paper establishes theoretical performance bounds for the greedy algorithm when maximizing non-submodular functions under constraints, especially under limited information scenarios, and validates these bounds with real-world autonomous underwater vehicle data.

Contribution

It provides the first theoretical bounds for greedy maximization of non-submodular functions with limited information, using curvature notions and real-world data validation.

Findings

The greedy algorithm achieves approximation guarantees despite non-submodularity.

Limited communication impacts performance but bounds still hold.

Real-world data confirms theoretical bounds are practical.

Abstract

We provide theoretical bounds on the worst case performance of the greedy algorithm in seeking to maximize a normalized, monotone, but not necessarily submodular objective function under a simple partition matroid constraint. We also provide worst case bounds on the performance of the greedy algorithm in the case that limited information is available at each planning step. We specifically consider limited information as a result of unreliable communications during distributed execution of the greedy algorithm. We utilize notions of curvature for normalized, monotone set functions to develop the bounds provided in this work. To demonstrate the value of the bounds provided in this work, we analyze a variant of the benefit of search objective function and show, using real-world data collected by an autonomous underwater vehicle, that theoretical approximation guarantees are achieved despite non-submodularity of the objective function.