# Faster Guarantees of Evolutionary Algorithms for Maximization of Monotone Submodular Functions

## TL;DR

This paper introduces new evolutionary algorithms with improved theoretical guarantees for maximizing monotone submodular functions under cardinality constraints, achieving near-optimal ratios with fewer function queries.

## Contribution

The paper proposes a novel Pareto optimization algorithm and a biased Pareto optimization variant with improved query complexity and approximation guarantees for submodular maximization.

## Key findings

- Algorithms achieve a $(1-rac{1}{e})$ approximation ratio in expectation.
- The algorithms require fewer function evaluations compared to existing methods.
- Empirical results support the theoretical improvements over stochastic greedy algorithms.

## Abstract

In this paper, the monotone submodular maximization problem (SM) is studied. SM is to find a subset of size $\kappa$ from a universe of size $n$ that maximizes a monotone submodular objective function $f$. We show using a novel analysis that the Pareto optimization algorithm achieves a worst-case ratio of $(1-\epsilon)(1-1/e)$ in expectation for every cardinality constraint $\kappa < P$, where $P\leq n+1$ is an input, in $O(nP\ln(1/\epsilon))$ queries of $f$. In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of $(1-\epsilon)(1-1/e)$ in expectation for every cardinality constraint $\kappa < P$ in $O(n\ln(P)\ln(1/\epsilon))$ queries of $f$. Further, the biased Pareto optimization algorithm can be modified in order to achieve a worst-case ratio of $(1-\epsilon)(1-1/e)$ in expectation for cardinality constraint $\kappa$ in $O(n\ln(1/\epsilon))$ queries of $f$. An empirical evaluation corroborates our theoretical analysis of the algorithms, as the algorithms exceed the stochastic greedy solution value at roughly when one would expect based upon our analysis.