1 research outputs found
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 coloring, discrepancy minimization, and interesting variants of
satisfiability. Similar to CSPs, the tractability of promise CSPs can be tied
to the structure of operations on the solution space called polymorphisms,
though in the promise world these operations are much less constrained. 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 Boolean promise CSPs when the constraint
predicates are symmetric. In this work, we vastly generalize these algorithmic
results. Specifically, we show that promise CSPs that admit a family of
"regional-periodic" polymorphisms are in P, 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.
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, 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
) to bypass ad-hoc adjustments for lying on a rounding
boundary.Comment: 41 pages, 2 figure