🤖 AI Summary
This paper investigates the computational complexity of 1-in-3 SAT and Not-All-Equal SAT under a “rainbow-free” structural weakening of the classical promise—where variable assignments must satisfy color constraints induced by a rainbow-free coloring. Employing an integrated approach combining constraint satisfaction theory, graph coloring and rainbow structure analysis, polynomial-time reductions, algebraic methods, and absorber techniques, we establish the first dichotomy theorem for this family of weakened-promise problems. The result shows that, except for trivial degenerate cases—namely, empty constraints or at most two variables—all nontrivial instances are strictly NP-hard; no intermediate complexity classes arise. This work reveals the decisive role of rainbow-free structure in CSP classification and fills a fundamental theoretical gap by providing a precise complexity characterization for weakened-promise variants of these canonical Boolean constraint satisfaction problems.
📝 Abstract
The 1-in-3 and Not-All-Eqal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Eqal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems.
