Sparsification of SAT and CSP Problems via Tractable Extensions

Abstract

Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for satisfiability (SAT) problems with a fixed constraint language Γ, every previously known result is captured by this approach; for several such problems, this is known to be tight. In this work, we investigate the limits of this approach—in particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint that can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism; using known algorithms for Maltsev languages, we can show that every problem of the latter type admits a “basis” of O ( n ) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G . For sparsifications of polynomial but super-linear size, we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a “lift-and-project” manner, by finding Maltsev extensions for constraints over c -tuples of variables, for a basis with O ( n c ) constraints. Additionally, we may use extensions with k -edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms φ 1 , φ 2 , …which characterizes whether a language Γ has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by φ 1 , the corresponding SAT problem does not admit a kernel of size O ( n 2−ε ) for any ε > 0 unless the polynomial hierarchy collapses.