🤖 AI Summary
This work investigates the tractability and computational hardness of Promise Constraint Satisfaction Problems (PCSPs) over Boolean domains. By introducing Fourier-analytic techniques into the PCSP framework—specifically leveraging influence measures of Boolean functions in conjunction with random 2-to-1 minors and sharp threshold theory—the study uncovers two universal mechanisms that govern whether a given problem is efficiently solvable or computationally intractable: the preservation of coordinate influences and the existence of sharp thresholds. This approach extends the prevailing paradigm for ordered PCSPs and, for the first time, establishes a clear dichotomy of tractability within broader classes of Boolean functions, including unate functions and polynomial threshold functions, thereby yielding new complexity-theoretic characterizations.
📝 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.