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On Boolean PCSPs with Polynomial Threshold Polymorphisms

Katzper Michno

TL;DR

The paper studies Boolean PCSPs whose polymorphisms are Polynomial Threshold Functions ($\mathsf{PTF}$) of bounded degree. It proves a dichotomy for $\mathsf{PTF}$s with non-negative coefficients ($\mathsf{PTF}_k^+$): the corresponding PCSP is either solvable in polynomial time or NP-hard, depending on a $k$-regularity condition and connections to symmetric/threshold function structure. For general $\mathsf{PTF}_k$, it shows NP-hardness under the Rich 2-to-1 Conjecture whenever there is a coordinate with significant influence in every polymorphism, leveraging a new influence-translation result for random $2$-to-$1$ minors under $p$-biased distributions. The key technical contributions include a probabilistic method to produce minor representations with small coordinate influence and a pull-back analysis showing how influence propagates through $p$-biased minor maps, enabling reductions from Gap Rich-2-to-1 Label Cover to PCSPs. Collectively, the work clarifies tractability borders for Boolean PCSPs with bounded-degree $\mathsf{PTF}$ polymorphisms and advances the use of Fourier-analytic tools in PCSP hardness reductions.

Abstract

In pursuit of a deeper understanding of Boolean Promise Constraint Satisfaction Problems (PCSPs), we identify a class of problems with restricted structural complexity, which could serve as a promising candidate for complete characterization. Specifically, we investigate the class of PCSPs whose polymorphisms are Polynomial Threshold Functions (PTFs) of bounded degree. We obtain two complexity characterization results: (1) with a hardness condition introduced in [ACMTCT'21], we establish a complete complexity dichotomy in the case where coefficients of PTF representations are non-negative; (2) dropping the non-negativity assumption, we show a hardness result for PTFs admitting coordinates with significant influence, conditioned on the Rich 2-to-1 Conjecture proposed in [ITCS'21]. In order to prove the latter, we show that a random 2-to-1 minor map retains significant coordinate influence over the $p$-biased hypercube with constant probability.

On Boolean PCSPs with Polynomial Threshold Polymorphisms

TL;DR

The paper studies Boolean PCSPs whose polymorphisms are Polynomial Threshold Functions () of bounded degree. It proves a dichotomy for s with non-negative coefficients (): the corresponding PCSP is either solvable in polynomial time or NP-hard, depending on a -regularity condition and connections to symmetric/threshold function structure. For general , it shows NP-hardness under the Rich 2-to-1 Conjecture whenever there is a coordinate with significant influence in every polymorphism, leveraging a new influence-translation result for random -to- minors under -biased distributions. The key technical contributions include a probabilistic method to produce minor representations with small coordinate influence and a pull-back analysis showing how influence propagates through -biased minor maps, enabling reductions from Gap Rich-2-to-1 Label Cover to PCSPs. Collectively, the work clarifies tractability borders for Boolean PCSPs with bounded-degree polymorphisms and advances the use of Fourier-analytic tools in PCSP hardness reductions.

Abstract

In pursuit of a deeper understanding of Boolean Promise Constraint Satisfaction Problems (PCSPs), we identify a class of problems with restricted structural complexity, which could serve as a promising candidate for complete characterization. Specifically, we investigate the class of PCSPs whose polymorphisms are Polynomial Threshold Functions (PTFs) of bounded degree. We obtain two complexity characterization results: (1) with a hardness condition introduced in [ACMTCT'21], we establish a complete complexity dichotomy in the case where coefficients of PTF representations are non-negative; (2) dropping the non-negativity assumption, we show a hardness result for PTFs admitting coordinates with significant influence, conditioned on the Rich 2-to-1 Conjecture proposed in [ITCS'21]. In order to prove the latter, we show that a random 2-to-1 minor map retains significant coordinate influence over the -biased hypercube with constant probability.

Paper Structure

This paper contains 4 sections, 4 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Constraint Satisfaction Problems and the algebraic approach
  3. Adding the promise to CSPs
  4. Fourier Analysis and Polynomial Threshold Functions

Key Result

Theorem 1.6

Suppose that $\mathbb{A}$ is a relational structure with finite universe. Then

Figures (1)

  • Figure 1: $K_3^2$ is a structure consisting of universe $[3]^2$ and a single symmetric, binary relation $\leftrightarrow$ defined as: $(a,b) \leftrightarrow (c,d)$ if and only if $a \leftrightarrow b$ and $c \leftrightarrow d$. It is simple to verify that the function $f: (x, y) \mapsto y$ is a homomorphism of $K_3^2$ to $K_3$. As a consequence, $f$ is a polymorphism of $K_3$.

Theorems & Definitions (12)

  • Example 1.1: $\mathsf{3}$-$\mathsf{SAT}$
  • Example 1.2: $\mathsf{3}$-$\mathsf{COLORING}$
  • Definition 1.3: Polymorphism
  • Definition 1.4: Minor
  • Theorem 1.6: CSP Dichotomy Theorem
  • Definition 1.8: Label Cover
  • Definition 1.9: Gap Label Cover
  • Theorem 1.11: Corollary 4.2 in layers
  • Conjecture 1.12: Rich 2-to-1 Conjecture
  • Theorem 1.14
  • ...and 2 more