Let $f$ be a $\{0,1\}$ -valued function over an integer $d$ -dimensional cube $\{0,1,\dots,n-1\}^d$ , for $n \geq 2$ and $d \geq 1$ . The function $f$ is called threshold if there exists a hyperplane which separates $0$ -valued points from $1$ -valued points. Let $C$ be a class of functions and $f \in C$ . A point $x$ is essential for the function $f$ with respect to $C$ if there exists a function $g \in C$ such that $x$ is a unique point on which $f$ differs from $g$ . A set of points $X$ is called teaching for the function $f$ with respect to $C$ if no function in $C \setminus \{f\}$ agrees with $f$ on $X$ . It is known that any threshold function has a unique minimal teaching set, which coincides with the set of its essential points. In this paper we study teaching sets of $k$ - threshold functions , i.e. functions that can be represented as a conjunction of $k$ threshold functions . We reveal a connection between essential points of $k$ threshold functions and essential points of the corresponding $k$ - threshold function . We note that, in general, a $k$ - threshold function is not specified by its essential points and can have more than one minimal teaching set. We show that for $d=2$ the number of minimal teaching sets for a 2- threshold function can grow as $Ω(n^2)$ . We also consider the class of polytopes with vertices in the $d$ -dimensional cube. Each polytope from this class can be defined by a $k$ - threshold function for some $k$ . In terms of $k$ - threshold functions we prove that a polytope with vertices in the $d$ -dimensional cube has a unique minimal teaching set which is equal to the set of its essential points. For $d=2$ we describe structure of the minimal teaching set of a polytope and show that cardinality of this set is either $Θ(n^2)$ or $O(n)$ and depends on the perimeter and the minimum angle of the polytope.
英文摘要：
Let $f$ be a $\{0,1\}$-valued function over an integer $d$-dimensional cube $\{0,1,\dots,n-1\}^d$, for $n \geq 2$ and $d \geq 1$. The function $f$ is called threshold if there exists a hyperplane which separates $0$-valued points from $1$-valued points. Let $C$ be a class of functions and $f \in C$. A point $x$ is essential for the function $f$ with respect to $C$ if there exists a function $g \in C$ such that $x$ is a unique point on which $f$ differs from $g$. A set of points $X$ is called teaching for the function $f$ with respect to $C$ if no function in $C \setminus \{f\}$ agrees with $f$ on $X$. It is known that any threshold function has a unique minimal teaching set, which coincides with the set of its essential points. In this paper we study teaching sets of $k$-threshold functions, i.e. functions that can be represented as a conjunction of $k$ threshold functions. We reveal a connection between essential points of $k$ threshold functions and essential points of the corresponding $k$-threshold function. We note that, in general, a $k$-threshold function is not specified by its essential points and can have more than one minimal teaching set. We show that for $d=2$ the number of minimal teaching sets for a 2-threshold function can grow as $Ω(n^2)$. We also consider the class of polytopes with vertices in the $d$-dimensional cube. Each polytope from this class can be defined by a $k$-threshold function for some $k$. In terms of $k$-threshold functions we prove that a polytope with vertices in the $d$-dimensional cube has a unique minimal teaching set which is equal to the set of its essential points. For $d=2$ we describe structure of the minimal teaching set of a polytope and show that cardinality of this set is either $Θ(n^2)$ or $O(n)$ and depends on the perimeter and the minimum angle of the polytope.
对于正整数$n$和$k$，若$n\geq 2k+1$为正整数，则Kneser图$K(n,k)$是由$\{1,\dots,n\}$的所有$k$ -集组成的顶点集的图，其中两个$k$ -集在不相交时恰好相邻. $K(n,k)$的独立集是$k$ -一致相交族，因此最大独立集由Erdens-Ko-Rado定理给出.设$K_p(n,k)$是$K(n,k)$的一个随机生成子图，其中每条边以概率$p$独立包含. Bollobás，Narayanan，and Raigorodskii问什么$p$ does $K_p(n,k)$ have the same independence number as $K(n,k)$ with high probability.对于$n=2k+1$，我们证明了一个命中时间的结果，这给了这个问题在$p=3/4$一个尖锐的阈值。此外，在完成Das和Tran以及Devlin和Kahn的工作的基础上，我们确定了所有$n>2k+1$的一个锐阈值函数。
在本文中，我们考虑的重建问题的树的硬核模型。我们确定了当lambda <（ln 2-o（1））ln^2（k）/（2 lnln（k））时k-正则树上的非重建机制的新界限，改进了lambda< e-1. This is almost tight as reconstruction is known to hold when lambda>（e+o（1））ln^2（k）的先前最佳界限。本文讨论了在稀疏随机图中寻找大独立集与树上独立集抽样的马尔可夫链的混合时间的关系。
