Maximization of Approximately Submodular Functions

TL;DR

This paper investigates the problem of maximizing functions that are approximately submodular within a cardinality constraint, providing bounds on query complexity and algorithms for different error levels.

Contribution

It characterizes the query complexity of maximizing approximately submodular functions and offers algorithms with constant approximation guarantees under certain conditions.

Findings

Exponential query complexity lower bound for psilon > n^{-1/2}

Constant approximation algorithms for psilon < 1/k

Analysis of bounds under bounded curvature assumption

Abstract

We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a submodular function break submodularity. Say that $F$ is $\varepsilon$-approximately submodular if there exists a submodular function $f$ such that $(1-\varepsilon)f(S) \leq F(S)\leq (1+\varepsilon)f(S)$ for all subsets $S$. We are interested in characterizing the query-complexity of maximizing $F$ subject to a cardinality constraint $k$ as a function of the error level $\varepsilon>0$. We provide both lower and upper bounds: for $\varepsilon>n^{-1/2}$ we show an exponential query-complexity lower bound. In contrast, when $\varepsilon< {1}/{k}$ or under a stronger bounded curvature assumption, we give constant approximation algorithms.