Beyond PCSP (1-in-3,NAE)

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 tractable. We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (t-in-k,NAE) by either adding tuples to t-in-k or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek, Guruswami, Wrochna, and \v{Z}ivn\'y [SICOMP'20]. For the latter, we classify all templates as either tractable or NP-hard.