Submodular Maximization through the Lens of Linear Programming

TL;DR

This paper establishes a connection between local optima in submodular maximization and approximation guarantees, introduces a general local search method applicable to various constraints, and shows linear optimization alone is insufficient for strong approximations.

Contribution

It extends the analogy of local optimality from linear programming to submodular maximization, introduces a unified local search framework, and answers an open question on the limitations of linear optimization oracles.

Findings

Local optima yield a 1/2-approximation for monotone submodular maximization.

A fast, general local search procedure applies to many constraint families.

Linear optimization oracles are insufficient for strong approximation guarantees.

Abstract

The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a $1/2$-approximation, and we also present an appropriate extension to the non-monotone setting. However, reaching a local optimum quickly is a non-trivial task. Moreover, we describe a fast and very general local search procedure that applies to a wide range of constraint families, and unifies as well as extends previous methods. In our framework, we match known approximation guarantees while disentangling and simplifying previous approaches. Moreover, despite its generality, we are able to show that our local search procedure is slightly faster than previous specialized methods. Furthermore, we resolve an open question on the relation between linear optimization and submodular maximization; namely, whether a linear optimization oracle may be enough to obtain strong approximation algorithms for submodular maximization. We show that this is not the case by providing an example of a constraint family on a ground set of size $n$ for which, if only given a linear optimization oracle, any algorithm for submodular maximization with a polynomial number of calls to the linear optimization oracle will have an approximation ratio of only $O ( \frac{1}{\sqrt{n}} \cdot \frac{\log n}{\log\log n} )$.