Fast and Private Submodular and $k$-Submodular Functions Maximization with Matroid Constraints

TL;DR

This paper develops fast, privacy-preserving algorithms for maximizing monotone submodular and $k$-submodular functions under matroid constraints, improving approximation guarantees and efficiency over previous methods.

Contribution

It introduces the first differentially private $(1-rac{1}{e})$-approximation for submodular maximization and a nearly linear-time $rac{1}{2}$-approximation for $k$-submodular maximization under matroid constraints.

Findings

Achieved a $(1-rac{1}{e})$-approximation with almost cubic function evaluations.

Provided the first differentially private $rac{1}{2}$-approximation for $k$-submodular maximization.

Improved approximation guarantees compared to previous algorithms.

Abstract

The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions. However, when such applications involve sensitive data about individuals, their privacy concerns should be addressed. In this paper, we study the problem of maximizing monotone submodular functions subject to matroid constraints in the framework of differential privacy. We provide $(1-\frac{1}{\mathrm{e}})$-approximation algorithm which improves upon the previous results in terms of approximation guarantee. This is done with an almost cubic number of function evaluations in our algorithm. Moreover, we study $k$-submodularity, a natural generalization of submodularity. We give the first $\frac{1}{2}$-approximation algorithm that preserves differential privacy for maximizing monotone $k$-submodular functions subject to matroid constraints. The approximation ratio is asymptotically tight and is obtained with an almost linear number of function evaluations.