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 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 problem admits a kernel of a particular size. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard 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 finite-domain CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP which is preserved by a Maltsev operation. This result utilise a variant of the simple algorithm for Maltsev constraints. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization for Boolean CSPs 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 $$\subseteq $$ co-NP/poly.