Submodular approximation: sampling-based algorithms and lower bounds

TL;DR

This paper explores the complexity of various submodular function optimization problems, providing new algorithms with approximation guarantees and establishing matching lower bounds to demonstrate their inherent difficulty.

Contribution

It introduces generalized submodular problems, offers approximation algorithms with sqrt(n/ln n) guarantees, and proves matching lower bounds, advancing understanding of submodular optimization complexity.

Findings

Approximation guarantees are of order sqrt(n/ln n).

Matching lower bounds confirm the inherent difficulty of these problems.

An improved lower bound for learning monotone submodular functions is provided.

Abstract

We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle. The approximation guarantees for most of our algorithms are of the order of sqrt(n/ln n). We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structure, whose guarantee is close to the lower bound. Although quite restrictive, the class of functions with this structure includes the ones that are used for lower bounds both by us and in previous work. This demonstrates that if there are significantly stronger lower bounds for this problem, they rely on more general submodular functions.