# Submodular Optimization Problems and Greedy Strategies: A Survey

## TL;DR

This survey reviews the effectiveness of greedy algorithms in submodular optimization problems, focusing on performance bounds and improvements for set and string submodular functions.

## Contribution

It provides a comprehensive overview of performance bounds for greedy strategies in submodular optimization, including recent improvements and bounds related to curvature and Nash equilibria.

## Key findings

- Performance bounds for greedy strategies are well-established.
- Improved bounds are available considering curvature of the objective.
- Batched greedy strategies and Nash equilibrium bounds are also analyzed.

## Abstract

The greedy strategy is an approximation algorithm to solve optimization problems arising in decision making with multiple actions. How good is the greedy strategy compared to the optimal solution? In this survey, we mainly consider two classes of optimization problems where the objective function is submodular. The first is set submodular optimization, which is to choose a set of actions to optimize a set submodular objective function, and the second is string submodular optimization, which is to choose an ordered set of actions to optimize a string submodular function. Our emphasis here is on performance bounds for the greedy strategy in submodular optimization problems. Specifically, we review performance bounds for the greedy strategy, more general and improved bounds in terms of curvature, performance bounds for the batched greedy strategy, and performance bounds for Nash equilibria.