Kernelization of Constraint Satisfaction Problems: A Study Through Universal Algebra

Abstract

A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the Boolean satisfiability problem (SAT), and the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determine whether a given SAT problem admits a kernel of a particular size. This could be contrasted to the currently flourishing research program of determining the classical complexity of finite-domain CSP problems, where almost all non-trivial tractable classes have been identified with the help of algebraic properties. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard SAT and CSP problems, parameterized by the number of variables. Our main focus is on problems admitting linear kernels, as has, somewhat surprisingly, previously been shown to exist. We show that a CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP problem which is preserved by a Maltsev operation. We also study extensions of this towards SAT and CSP problems with kernels with O(n c ) constraints, c > 1, based on embeddings into CSP problems preserved by a k-edge operation, k ≤ c + 1. These results follow via a variant of the celebrated few subpowers algorithm. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization of SAT problems with linear kernels, by showing that an algebraic condition that is shared by all problems with a Maltsev embedding is also necessary for the existence of a linear kernel unless NP ⊆ co-NP/poly.

Best-Case and Worst-Case Sparsifiability of Boolean CSPs

“…Furthermore, we fully classified the sparsifiability of CSP(Γ) when Γ contains relations of arity at most three, based on the arity of the largest or that can be cone-defined from Γ. It follows from results of Lagerkvist and Wahlström [17] that for constraint languages of arbitrary arity, the exponent of the best sparsification size does not always match the arity of the largest or cone-definable from Γ. (This will be described in more detail in the upcoming journal version of this work.)…”

“…We conclude with a brief discussion on the relation between our polynomial-based framework for linear compression and the framework of Lagerkvist and Wahlström [17]. They used a different method for sparsification, based on embedding a Boolean constraint language Γ into a constraint language Γ defined over a larger domain D, such that Γ is preserved by a Maltsev operation.…”

“…A key result in this area is Schaefer's dichotomy theorem [20], which classifies each CSP over the Boolean domain as polynomial-time solvable or NP-complete. Continuing a recent line of investigation [12,14,17], we aim to understand for which NP-complete CSPs an instance can be sparsified in polynomial time, without changing the answer. In particular, we investigate the following questions.…”

“…The polynomial-based framework [12] also resulted in some linear sparsifications. Since "1-in-d" constraints (to satisfy a clause, exactly one out of its ≤ d literals should evaluate to true) can be captured by linear polynomials, the 1-in-d-SAT problem has a sparsification with O(n) constraints for each constant d. This prompted a detailed investigation into linear sparsifications for CSPs by Lagerkvist and Wahlström [17], who used the toolkit of universal algebra in an attempt to obtain a characterization of the Boolean CSPs with a linear sparsification. Their results give a necessary and sufficient condition on the constraint language of a CSP for having a so-called Maltsev embedding over an infinite domain.…”

“…By Proposition A.3, we obtain that Γ * pp-defines all Boolean relations. It follows from [17,Lemma 17] that Γ * pp-defines H n,k using O(n + k) constraints and O(n + k) existentially quantified variables. We add the constraints from this pp-definition to C, and add the existentially quantification variables to V .…”