更快更简单的估值函数草图
Faster and Simpler Sketches of Valuation Functions
摘要：
我们提出了快速算法绘制赋值函数。设$N$（$|N|=n$）是某个基集，$v:2^N\rightarrow \mathbb R$是一个函数。我们说$\tilde v:2^N\rightarrow \mathbb R$是$v$的$α$ -草图，如果对于每个$S$集合，我们有$\frac {v(S)} α \leq \tilde v(S) \leq v(S)$和$\tilde v$可以用$poly(n)$位来描述。Goemans等人。[SODA'09]表明，如果$v$是次模的，那么存在一个$\tilde O(\sqrt n)$ -草图，可以使用多项式许多值查询来构造（这是最好的可能性，因为Balcan和Harvey [STOC'11]表明没有次模函数允许$n^{\frac 1 3 - ε}$ -草图）。基于他们的工作，Balcan等人[COLT'12]和Badanidiyuru等人[SODA'12]表明，如果$v$是次可加的，则存在可以使用多项式许多需求查询来构造的$\tilde O(\sqrt n)$草图。所有以前的草图都是基于复杂的几何结构。在他们的建设的第一步是证明存在一个良好的草图通过找到一个椭圆，"近似“$v$好（这是通过应用约翰的定理，以确保存在一个椭圆是"接近”与$v$相关联的多拟阵）。第二步是证明这个椭球可以有效地找到，这是通过反复求解某个凸规划来获得约翰椭球的更好近似来完成的。在本文中，我们给出了一个更简单的，非几何证明的存在良好的草图，并利用证明，以获得更快的算法，匹配以前获得的近似界。具体来说，我们提供了一个算法，发现$\tilde O(\sqrt n)$ -草图的子模块功能只有$\tilde O(n^\frac{3}{2})$值查询，和算法，发现$\tilde O(\sqrt n)$ -草图的子加性功能与$O(n)$的需求和值查询。
英文摘要：
We present fast algorithms for sketching valuation functions. Let $N$ ($|N|=n$) be some ground set and $v:2^N\rightarrow \mathbb R$ be a function. We say that $\tilde v:2^N\rightarrow \mathbb R$ is an $α$-sketch of $v$ if for every set $S$ we have that $\frac {v(S)} α \leq \tilde v(S) \leq v(S)$ and $\tilde v$ can be described in $poly(n)$ bits. Goemans et al. [SODA'09] showed that if $v$ is submodular then there exists an $\tilde O(\sqrt n)$-sketch that can be constructed using polynomially many value queries (this is the best possible, as Balcan and Harvey [STOC'11] show that no submodular function admit an $n^{\frac 1 3 - ε}$-sketch). Based on their work, Balcan et al. [COLT'12] and Badanidiyuru et al. [SODA'12] show that if $v$ is subadditive then there exists an $\tilde O(\sqrt n)$-sketch that can be constructed using polynomially many demand queries. All previous sketches are based on complicated geometric constructions. The first step in their constructions is proving the existence of a good sketch by finding an ellipsoid that ``approximates'' $v$ well (this is done by applying John's theorem to ensure the existence of an ellipsoid that is ``close'' to the polymatroid that is associated with $v$). The second step is showing this ellipsoid can be found efficiently, and this is done by repeatedly solving a certain convex program to obtain better approximations of John's ellipsoid. In this paper, we give a much simpler, non-geometric proof for the existence of good sketches, and utilize the proof to obtain much faster algorithms that match the previously obtained approximation bounds. Specifically, we provide an algorithm that finds $\tilde O(\sqrt n)$-sketch of a submodular function with only $\tilde O(n^\frac{3}{2})$ value queries, and an algorithm that finds $\tilde O(\sqrt n)$-sketch of a subadditive function with $O(n)$ demand and value queries.
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