Boolean PCSPs through the lens of Fourier Analysis
Demian Banakh, Katzper Michno
TL;DR
This paper introduces a Fourier analysis-based framework for Boolean PCSPs, revealing phenomena linked to hardness and tractability, and extends prior work to broader classes of functions.
Contribution
It generalizes the analysis of Boolean PCSPs by identifying influence preservation and sharp thresholds as indicators of complexity, expanding previous results.
Findings
Influence preservation under random minors correlates with problem hardness.
Presence of sharp thresholds indicates tractability.
Broader classes of functions like unate and polynomial threshold functions are analyzed.
Abstract
We develop an analytical framework for Boolean Promise Constraint Satisfaction Problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek, Guruswami, and Sandeep [ICALP'21] on Ordered PCSPs, we identify two general phenomena in Boolean minions indicative of hardness or tractability: (1) preservation of coordinate influence under random 2-to-1 minors and (2) the presence of sharp thresholds. We demonstrate that these phenomena occur in broader settings than previously established, yielding new hardness/tractability results for minions consisting of unate or polynomial threshold functions.
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