Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: $(1)$ reduce the enumeration in the tight $(1-e^{-1})$-approximation of [Sviridenko 04] from subsets of size three to two; $(2)$ present an improved upper bound of $0.42945$ for the classic algorithm which returns the better between a single element and the output of the greedy heuristic.
翻译:在受Knapsack限制的情况下最大限度地增加单调子模量功能的许多算法都依赖于自然贪婪的脂质。我们对这一贪婪的脂质进行了新的精细分析,从而使我们能够:1美元减少(1-e ⁇ - ⁇ -1})美元(Sviridenko 04)这一紧凑的三至二级的[Sviridenko 04]的查点;2美元为经典算法提供了4,2945美元的改进上限,该算法在单一元素和贪婪的脂质输出之间回流得更好。