发表机构
Nanyang Technological University; University of Technology Sydney; Great Bay University（南洋理工大学; 悉尼科技大学; 大湾区大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对拟阵约束下γ-弱子模函数最大化问题，提出基于非齐次泊松时钟的MGPE算法，实现优于现有界且渐近最优的近似比，并自动恢复特殊情形下的紧比率。
AI 中文摘要
在过去十年中，越来越多的研究表明，γ-弱子模性广泛出现在众多子集选择任务中，包括特征选择、神经网络剪枝和视频摘要。尽管其普遍存在，但在一般拟阵约束下最大化γ-弱子模函数仍然具有挑战性。迄今为止，唯一已知的近似保证是由chen2018weakly建立的保守因子(1+1/γ)^{-2}。为了改进这一结果，本文提出了一种名为MGPE的新算法，该算法通过仔细控制非齐次泊松时钟，反复执行最大增益局部交换，并证明了MGPE可以达到任意接近ρ_γ=1-(γ/(2-γ))^{γ^2/(2(1-γ))}的近似比。与之前的保证形成鲜明对比的是，我们获得的因子ρ_γ不仅对于每个γ∈(0,1]严格优于(1+1/γ)^{-2}，而且当γ→1时可以渐近逼近子模最大化的最优(1-1/e)近似。此外，我们惊奇地发现，当拟阵约束简化为基数约束或目标满足更强的α-弱DR子模性时，MGPE可以自动恢复紧的近似比1-e^{-γ}和1-e^{-α}。这里，α∈(0,1]表示DR比。
英文摘要
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.
Comments55 pages
