On the Approximability of Presidential Type Predicates

Neng Huang, Aaron Potechin, Jarosław Byrka, Raghu Meka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Given a predicate P: {-1, 1}^k → {-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
more » ... 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 δ₀ > 0, there exists a k₀ such that if k ≥ k₀, δ ∈ (δ₀,1 - 2/k], and {δ}k + k - 1 is an odd integer then the presidential type predicate P(x) = sign({δ}k{x₁} + ∑_{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.
doi:10.4230/lipics.approx/random.2020.58 dblp:conf/approx/HuangP20 fatcat:3j4bamxcgfcbzh4uv4yixjotxi