AI 中文总结
本文针对拟阵独立性约束下的次模最大化问题，提出一种恶意泊松过程混合算法，适用于非单调与单调次模函数，分别达到1/e和1-1/e的近似比，还得到相关快速近似算法。
AI 中文摘要
我们研究在拟阵独立性约束下最大化一般的、不一定单调的次模函数的问题。该问题历史悠久，已有多种使用离散和连续方法的算法。最近，[Ganz-Rozenman、Kulik、Schwartz和Singh STOC `26]针对次模函数为单调的该问题的特殊情况，提出了一种基于泊松过程的新颖混合方法，旨在结合离散和连续方法的优势。我们的主要结果是一种新的基于泊松过程的混合算法，该算法适用于非单调和单调次模函数，对前者实现1/e的近似比，对后者实现1-1/e的近似比。该算法始终保持一个可行集，并在泊松过程控制的随机时刻，基于最优响应集执行单个元素的交换。新的思路是我们的算法是恶意的，因为它可以故意舍弃同时属于当前集和最优响应集的元素。令人惊讶的是，这种恶意步骤不会损害算法对单调次模函数的近似效果，但对非单调情况是必要的。作为应用，我们获得了在一般拟阵独立性约束下最大化非单调次模函数的快速近似算法，以及针对划分拟阵的更快算法。
英文摘要
We study the problem of maximizing a general and not necessarily monotone submodular function subject to a matroid independence constraint. This problem has a rich history, with multiple algorithms using both discrete and continuous methods. Recently, [Ganz-Rozenman, Kulik, Schwartz and Singh STOC `26] presented a novel hybrid approach based on a Poisson process that aims to combine the strengths of both discrete and continuous methods for the special case of the problem where the submodular function is monotone. Our main result is a new Poisson process based hybrid algorithm that works for both non-monotone and monotone submodular functions, achieving an approximation of $ \frac{1}{e}$ for the former and $1-\frac{1}{e}$ for the latter. The algorithm always maintains a feasible set and at random times governed by the Poisson process it performs a single element swap based on a best response set. The new idea is that our algorithm is spiteful as it can purposefully discard an element that is in both the current set and the best response set. Surprisingly, this spiteful step does not harm the approximation our algorithm achieves for monotone submodular functions but is necessary for the non-monotone case. As applications, we obtain fast approximation algorithms for maximizing non-monotone submodular function subject to a general matroid independence constraint as well as faster algorithms for a partition matroid.
