Promises Make Finite (Constraint Satisfaction) Problems Infinitary

TL;DR

This paper demonstrates that certain Promise Constraint Satisfaction Problems (PCSPs) over finite domains inherently require infinite domains for their tractable reductions, highlighting fundamental complexity limitations.

Contribution

It proves that a specific finite-domain PCSP cannot be reduced to a tractable finite-domain CSP unless P=NP, establishing a key complexity barrier.

Findings

Finite-domain PCSPs may require infinite domains for tractable reductions

The (1-in-3-SAT, Not-All-Equal-3-SAT) PCSP cannot be simplified to finite-domain CSPs

Complexity barriers are identified for certain PCSP reductions

Abstract

The fixed template Promise Constraint Satisfaction Problem (PCSP) is a recently proposed significant generalization of the fixed template CSP, which includes approximation variants of satisfiability and graph coloring problems. All the currently known tractable (i.e., solvable in polynomial time) PCSPs over finite templates can be reduced, in a certain natural way, to tractable CSPs. However, such CSPs are often over infinite domains. We show that the infinity is in fact necessary by proving that a specific finite-domain PCSP, namely (1-in-3-SAT, Not-All-Equal-3-SAT), cannot be naturally reduced to a tractable finite-domain CSP, unless P=NP.