# Constrained Submodular Maximization via Greedy Local Search

## TL;DR

This paper introduces a simple greedy local search algorithm that achieves near-optimal approximation ratios for maximizing monotone submodular functions under complex constraints like knapsacks and multiple matroids.

## Contribution

The paper presents a new combinatorial algorithm combining greedy and local search for submodular maximization with knapsack and matroid constraints, extending known approximation guarantees.

## Key findings

- Achieves a $rac{1 - e^{-2}}{2}$-approximation for knapsack and matroid constraints.
- Extends to a $rac{1 - e^{-(k+1)}}{k+1}$ ratio for multiple matroid intersections.
- Demonstrates the effectiveness of combining greedy algorithms with local search in constrained submodular maximization.

## Abstract

We present a simple combinatorial $\frac{1 -e^{-2}}{2}$-approximation algorithm for maximizing a monotone submodular function subject to a knapsack and a matroid constraint. This classic problem is known to be hard to approximate within factor better than $1 - 1/e$. We show that the algorithm can be extended to yield a ratio of $\frac{1 - e^{-(k+1)}}{k+1}$ for the problem with a single knapsack and the intersection of $k$ matroid constraints, for any fixed $k > 1$. Our algorithms, which combine the greedy algorithm of [Khuller, Moss and Naor, 1999] and [Sviridenko, 2004] with local search, show the power of this natural framework in submodular maximization with combined constraints.