On the Approximability of Presidential Type Predicates

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 $\delta_0 > 0$, there exists a $k_0$ such that if $k \geq k_0$, $\delta \in (\delta_0,1 - 2/k]$, and ${\delta}k + k - 1$ is an odd integer then the presidential type predicate $P(x) = sign({\delta}k{x_1} + \sum_{i=2}^{k}{x_i})$ 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.