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arXiv · 2411.10949

Maximization of Approximately Submodular Functions

Thibaut Horel·Yaron Singer
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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.

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Thibaut Horel, Yaron Singer. 2024-11-17. Maximization of Approximately Submodular Functions. https://arxiv.org/abs/2411.10949

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