Sensitivity Analysis of Submodular Function Maximization

TL;DR

This paper investigates the sensitivity of algorithms for monotone submodular maximization, revealing limitations on sensitivity in general but identifying conditions for low sensitivity when the constraint is large.

Contribution

It provides a theoretical analysis of worst-case sensitivity, establishing bounds and conditions under which low sensitivity can be achieved.

Findings

Non-trivial sensitivity of o(k) is not possible for many algorithms

Low curvature enables O(1) sensitivity when k = Ω(n)

Results hold even in distributed settings

Abstract

We study the recently introduced idea of worst-case sensitivity for monotone submodular maximization with cardinality constraint $k$, which captures the degree to which the output argument changes on deletion of an element in the input. We find that for large classes of algorithms that non-trivial sensitivity of $o(k)$ is not possible, even with bounded curvature, and that these results also hold in the distributed framework. However, we also show that in the regime $k = \Omega(n)$ that we can obtain $O(1)$ sensitivity for sufficiently low curvature.