# Approximate Submodularity and Its Implications in Discrete Optimization

## TL;DR

This paper introduces metrics for approximate submodularity, extending key properties and analytical tools from submodular optimization to broader contexts in discrete optimization, including mixed-integer sets.

## Contribution

It defines new metrics for approximate submodularity and generalizes existing analytical tools to this broader setting in discrete optimization.

## Key findings

- Metrics for approximate submodularity are introduced.
- Properties of approximate submodularity preservation are derived.
- Extensions of submodular optimization tools to approximate cases are demonstrated.

## Abstract

Submodularity is a key property in discrete optimization. Submodularity has been widely used for analyzing the greedy algorithm to give performance bounds and providing insight into the construction of valid inequalities for mixed-integer programs. In recent years, researchers started to study approximate submodularity, with a primary focus on providing performance bounds for iterative approaches. In this paper, we study approximate submodularity from a different perspective in order to broaden its use cases in discrete optimization. We define metrics that quantify approximate submodularity, which we then use to derive new properties about both approximate submodularity preservation and the well-known Lov\'{a}sz extension for set functions. We also show that previous analyses of mixed-integer sets, such as the submodular knapsack polytope, can be extended to the approximate submodularity setting. Our work demonstrates that one may generalize many of the analytical tools used in submodular optimization into the approximate submodularity context.