Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

2026-09-21

AI Summary

本文提出了一种新的算法MGPE，通过非均匀泊松时钟控制下的最大增益局部交换来解决γ-弱次模函数在一般拟阵约束下的最大化问题，提供了更优的近似比。

Abstract

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

Problem

• subset selection
• matroid constraint
• weak submodularity
• approximation algorithm
• optimization

Innovation

• Poisson Clock
• Maximum-Gain Local Exchanges
• Weak Submodularity
• Matroid Constraint
• Approximation Ratio