关于总统型谓词的可逼近性
On the Approximability of Presidential Type Predicates
摘要：
给定谓词$P: \{-1, 1\}^k \to \{-1, 1\}$，设$CSP(P)$是约束满足问题的集合，其约束具有$P$的形式。我们说$P$是可近似的，如果给定$CSP(P)$的一个几乎可满足的实例，存在一个比随机分配更好的概率多项式时间算法。否则，我们说$P$是近似抵抗的。在本文中，我们分析了总统类型谓词，这是平衡的线性阈值函数，除了第一个变量（总统）的所有变量具有相同的权重。我们证明了几乎所有的总统型谓词$P$都是可近似的。更精确地说，我们证明了以下结果：对于任何$δ_0 > 0$，存在一个$k_0$，使得如果$k \geq k_0$，$δ\in (δ_0,1 - 2/k]$和$δk + k - 1$是一个奇数，那么总统类型谓词$P(x) = sign(δk{x_1} + \sum_{i=2}^{k}{x_i})$是可逼近的。为了证明这一点，我们构建了一个舍入方案，利用偏差和成对偏差。我们还提供证据表明，使用成对偏差是必要的，这样的舍入方案。
英文摘要：
Given a predicate $P: \{-1, 1\}^k \to \{-1, 1\}$, let $CSP(P)$ be the set of constraint satisfaction problems whose constraints are of the form $P$. We say that $P$ is approximable if given a nearly satisfiable instance of $CSP(P)$, there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that $P$ is approximation resistant. In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential-type predicates $P$ are approximable. More precisely, we prove the following result: for any $δ_0 > 0$, there exists a $k_0$ such that if $k \geq k_0$, $δ\in (δ_0,1 - 2/k]$, and $δk + k - 1$ is an odd integer then the presidential type predicate $P(x) = sign(δk{x_1} + \sum_{i=2}^{k}{x_i})$ is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes.
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