Boolean PCSPs through the lens of Fourier Analysis
It advances the theoretical understanding of Boolean PCSPs by providing new analytical tools and results, though the scope is domain-specific to constraint satisfaction.
The paper develops a Fourier-analytic framework for Boolean Promise Constraint Satisfaction Problems (PCSPs), identifying two general phenomena in Boolean minions that indicate hardness or tractability, and yields new results for minions of unate or polynomial threshold functions.
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.