Maximizing a submodular function is a fundamental task in machine learning
and in this paper we study the deletion robust version of the problem under the
classic matroids constraint. Here the goal is to extract a small size summary
of the dataset that contains a high value independent set even after an
adversary deleted some elements. We present constant-factor approximation
algorithms, whose space complexity depends on the rank $k$ of the matroid and
the number $d$ of deleted elements. In the centralized setting we present a
$(4.597+O(\varepsilon))$-approximation algorithm with summary size $O(
\frac{k+d}{\varepsilon^2}\log \frac{k}{\varepsilon})$ that is improved to a
$(3.582+O(\varepsilon))$-approximation with $O(k + \frac{d}{\varepsilon^2}\log
\frac{k}{\varepsilon})$ summary size when the objective is monotone. In the
streaming setting we provide a $(9.435 + O(\varepsilon))$-approximation
algorithm with summary size and memory $O(k + \frac{d}{\varepsilon^2}\log
\frac{k}{\varepsilon})$; the approximation factor is then improved to
$(5.582+O(\varepsilon))$ in the monotone case.

该研究探讨基于经典拟阵约束下的删除鲁棒版本的子模函数最大化问题，在此问题中，目标是提取一个小型数据集的摘要，即使在对手删除一些元素后也包含高价值独立集，该研究提供了具有常数因子逼近比的近似算法，其中空间复杂度取决于拟阵的秩 k 和已删除元素的数量 d。

基于单调性不稳定的麦特罗伊德上的非单调子模最大化

Deletion Robust Non-Monotone Submodular Maximization over Matroids

Maximizing a monotone submodular function is a fundamental task in machine
learning. In this paper, we study the deletion robust version of the problem
under the classic matroids constraint. Here the goal is to extract a small size
summary of the dataset that contains a high value independent set even after an
adversary deleted some elements. We present constant-factor approximation
algorithms, whose space complexity depends on the rank $k$ of the matroid and
the number $d$ of deleted elements. In the centralized setting we present a
$(3.582+O(\varepsilon))$-approximation algorithm with summary size $O(k +
\frac{d \log k}{\varepsilon^2})$. In the streaming setting we provide a
$(5.582+O(\varepsilon))$-approximation algorithm with summary size and memory
$O(k + \frac{d \log k}{\varepsilon^2})$. We complement our theoretical results
with an in-depth experimental analysis showing the effectiveness of our
algorithms on real-world datasets.

本文研究在传统的拟阵约束下，单调子模函数的删除鲁棒版本的最大化问题，提出了空间复杂度依赖于拟阵的秩和删除元素数量的常数因子逼近算法，在中心化、流式环境下均获得了好的结果。