Symmetric Polymorphisms and Efficient Decidability of Promise CSPs
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In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem–which has recently benefited from multiple breakthroughs [BKO19, WZ19] due to a systematic study of promise CSPs under the lens of "polymorphisms," operations that map tuples in the strict form of each constraint to a tuple in its weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, that is the coordinates are permutation invariant. This generalizes previous work of the authors [BG19]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic theorem and shed further light on how symmetries of polymorphisms enable algorithms.
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