# Deterministic & Adaptive Non-Submodular Maximization via the Primal Curvature

## TL;DR

This paper introduces a new technique for analyzing the performance of greedy algorithms in maximizing non-submodular functions, extending classical guarantees to adaptive and stochastic settings.

## Contribution

It presents a novel curvature-based method to bound greedy algorithm performance for non-submodular functions, including adaptive and stochastic cases.

## Key findings

- Provides a curvature-based approximation ratio for non-submodular maximization
- Extends classical ratios to adaptive greedy algorithms
- Supports applications with incomplete data and uncertainty

## Abstract

While greedy algorithms have long been observed to perform well on a wide variety of problems, up to now approximation ratios have only been known for their application to problems having submodular objective functions $f$. Since many practical problems have non-submodular $f$, there is a critical need to devise new techniques to bound the performance of greedy algorithms in the case of non-submodularity. Our primary contribution is the introduction of a novel technique for estimating the approximation ratio of the greedy algorithm for maximization of monotone non-decreasing functions based on the curvature of $f$ without relying on the submodularity constraint. We show that this technique reduces to the classical $(1 - 1/e)$ ratio for submodular functions. Furthermore, we develop an extension of this ratio to the adaptive greedy algorithm, which allows applications to non-submodular stochastic maximization problems. This notably extends support to applications modeling incomplete data with uncertainty.