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 SAT problems with a fixed constraint language Γ, every previously known result is captured by this approach, and 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 which 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, and 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 φ1, φ2,. .. 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 φ1, the corresponding SAT problem does not admit a kernel of size O(n 2−ε) for any ε > 0 unless the polynomial hierarchy collapses.

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

“…Proof. This is a special case of the notion of a Maltsev embedding of 𝑅 previously investigated by the authors [42].…”

“…Another problem is to find a generalisation of the algorithm for constraints defined via bounded-degree polynomials [48], without explicitly using properties specific to polynomials. A different generalisation of this class was considered by the present authors in the form of relations with bounded-degree Maltsev embeddings [42]. Since this properly generalises bounded-degree polynomials, it is natural to ask whether this class admits an improved algorithm.…”

“…Although there is no formal technical connection, variations on partial Maltsev polymorphisms occur as a possible success criterion for the latter. However, unlike the case of improved running times, it is known that no finite set of partial polymorphisms can suffice for polynomial sparsification [42]. Thus, in our earlier work [41,42] we studied properties of invariant sets Inv(𝐹 ), and properties of certain specific partial operations, but were unable to explicitly make the connection between these types of operations and improved running times.…”

“…Another related topic is the investigation of polynomial (or linear) kernels for SAT(Γ) parameterized by 𝑛, also known as sparsification. The best known results are produced by phrasing constraints as roots of bounded-degree polynomials evaluated at {0, 1} 𝑛 [16], with an extension in terms of Maltsev languages which may or may not be more powerful [42].…”

“…This follows from the fact that systems of such polynomials have bounded rank; hence any qfpp-definition over this language only needs to use 1) polynomials. Since there are at most 2 𝑛 𝑂 (1) distinct polynomials once the degree and the (finite) field have been fixed, the sparseness follows (see Jansen & Pieterse [34] and Lagerkvist & Wahlström [42] for extensions and related sparsification applications). With this in mind, it would seem that a complete and explicit characterisation of relations in Inv(𝑓 ) is not likely to be useful.…”