Beyond PCSP (1-in-3,NAE)
Alex Brandts, Stanislav \v{Z}ivn\'y

TL;DR
This paper investigates non-symmetric promise CSPs derived from Boolean templates, classifying their computational complexity and identifying tractable cases beyond known symmetric scenarios.
Contribution
It extends the understanding of PCSP complexity by classifying non-symmetric Boolean templates, especially those obtained by adding or removing tuples from standard templates.
Findings
Classified templates as either tractable or NP-hard.
Identified cases solvable by the strongest known algorithms.
Extended complexity results beyond symmetric PCSPs.
Abstract
The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs. 1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SICOMP'21] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (1-in-3,NAE) is…
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