An Algorithmic Blend of LPs and Ring Equations for Promise CSPs.

Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well known problems can be cast as promise CSPs including approximate graph and hypergraph coloring, discrepancy minimization, and interesting variants of satisfiability. Similar to CSPs, the tractability of promise CSPs can be tied to the structure of associated operations on the solution space called (weak) polymorphisms. However, compared to CSPs whose polymorphisms are well-structured algebraic objects called clones, polymorphisms in the promise world are much less constrained --- essentially any infinite family of functions obeying mild conditions can arise as polymorphisms. Under the thesis that non-trivial polymorphisms govern tractability, promise CSPs therefore provide a fertile ground for the discovery of novel algorithms. In previous work, we classified all tractable cases of Boolean promise CSPs when the constraint predicates are symmetric. The algorithms were governed by three kinds of polymorphism families: (i) parity functions, (ii) majority functions, or (iii) a non-symmetric (albeit block-symmetric) family we called alternating threshold. In this work, we provide a vast generalization of these algorithmic results. Specifically, we show that promise CSPs that admit a family of "regional-periodic" polymorphisms are solvable in polynomial time, assuming that determining which region a point is in can be computed in polynomial time. Such polymorphisms are quite general and are obtained by gluing together several functions that are periodic in the Hamming weights in different blocks of the input. For example, we can have functions that equal parity for relative Hamming weights up to 1/2, and Majority (so identically 1) for weights above 1/2. Our algorithm is based on a novel combination of linear programming and solving linear systems over rings. We also abstract a framework based on reducing a promise CSP to a CSP over an infinite domain, solving it there (via the said combination of LPs and ring equations), and then rounding the solution to an assignment for the promise CSP instance. The rounding step is intimately tied to the family of polymorphisms, and clarifies the connection between polymorphisms and algorithms in this context. As a key ingredient, we introduce the technique of finding a solution to a linear program with integer coefficients that lies in a different ring (such as [MATH HERE]) to bypass ad-hoc adjustments for lying on a rounding boundary.