[Paper Review] On the Approximability of Presidential Type Predicates

Neng Huang, Aaron Potechin|arXiv (Cornell University)|Jul 9, 2019

Complexity and Algorithms in Graphs13 references2 citations

TL;DR

This paper establishes that almost all presidential-type predicates—balanced linear threshold functions where the 'president' variable has significantly higher weight than others—are approximable via a novel rounding scheme leveraging both individual and pairwise biases in semidefinite programming. The key result shows that for sufficiently large arity k and weight δk with δ ∈ (δ₀, 1 − 2/k], such predicates admit a constant-factor approximation, demonstrating that pairwise biases are essential for achieving this approximation beyond low-degree schemes.

ABSTRACT

Given a predicate $P: \{-1, 1\}^k o \{-1, 1\}$, 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.

Motivation & Objective

• To determine the approximability of presidential-type predicates, a class of balanced linear threshold functions where one variable (the 'president') has higher weight.
• To investigate whether such predicates can be approximated better than random assignment using polynomial-time algorithms.
• To analyze the necessity of pairwise biases in rounding schemes for achieving non-trivial approximation ratios.
• To extend prior inapproximability results by identifying conditions under which presidential-type predicates become approximable.
• To explore the structural limitations of low-degree rounding schemes that rely only on individual biases.

Proposed method

• Constructs a rounding scheme based on biases {bi} and pairwise biases {bij} derived from a standard semidefinite programming relaxation.
• Uses Fourier analysis of the predicate to compare the magnitude of the president's Fourier coefficient ˆPP with those of the citizens ˆPC, showing ˆPP is exponentially larger.
• Designs a mixed strategy distribution µ over points in the KTW polytope such that all degree-a moments (for a ≤ m) vanish, ensuring no bias toward over- or under-satisfaction.
• Employs a probabilistic construction of points in the KTW polytope with specific sign patterns to balance contributions across degrees 1, 3, 4, and 5, using exponentially small probabilities for high-impact terms.
• Demonstrates that for fixed m, no rounding scheme of degree ≤ m can succeed without using pairwise biases, by showing that such schemes fail to balance higher-degree moments.
• Applies the KTW polytope framework to model feasible bias and pairwise bias vectors, ensuring consistency with the SDP relaxation.

Experimental results

Research questions

• RQ1 For which presidential-type predicates is a constant-factor approximation possible using polynomial-time algorithms?
• RQ2 Is the use of pairwise biases necessary for achieving non-trivial approximation in rounding schemes for these predicates?
• RQ3 Can low-degree rounding schemes (with bounded degree m) succeed without using pairwise biases, or is their inclusion essential?
• RQ4 How does the relative weight of the 'president' variable (δk) affect the approximability of the predicate?
• RQ5 Are there structural limitations in the KTW polytope that prevent certain rounding schemes, especially for predicates with strong symmetry or dominance?

Key findings

• For any δ₀ > 0, there exists k₀ such that for all k ≥ k₀ and δ ∈ (δ₀, 1 − 2/k] with δk + k − 1 odd, the predicate P(x) = sign(δk x₁ + ∑_{i=2}^k x_i) is approximable.
• The rounding scheme achieves a constant advantage over random assignment by exploiting both individual and pairwise biases in the SDP solution.
• The construction shows that pairwise biases are necessary: no fixed-degree rounding scheme without them can achieve approximation for large k.
• The Fourier coefficient of the president variable ˆPP is exponentially larger than that of the citizens ˆPC, which enables the balancing of higher-degree moments through strategic probability allocation.
• For m = 5, a mixed strategy distribution µ is explicitly constructed with five point types, using exponentially small probabilities for the fifth type to balance high-degree terms.
• The method fails for the monarchy predicate P(x) = sign((k−2)x₁ + ∑_{i=2}^k x_i) due to geometric constraints in the KTW polytope, such as bi ≥ −b₁ for i ≥ 2.