🤖 AI Summary
This work investigates the finite tractability of Promise Constraint Satisfaction Problems (PCSPs): specifically, when a PCSP admits a polynomial-time reduction to an efficiently solvable CSP over some finite domain (unless P = NP). Employing algebraic methods—including clone theory, polymorphism analysis, and categorical tools—alongside structural properties of templates such as symmetry and negation closure, we establish the first necessary and sufficient condition for finite tractability within the class of symmetric Boolean PCSP templates. This resolves a fundamental gap left open by prior work of Barto and Brakensiek–Guruswami. Our characterization shows that most classical approximation problems—including 1-in-3-SAT, Not-All-Equal SAT, and various approximate graph coloring variants—are not finitely tractable. Moreover, we provide a complete classification boundary delineating precisely which symmetric Boolean PCSPs are finitely tractable.
📝 Abstract
The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].