Kernelization of Constraint Satisfaction Problems: A Study through Universal Algebra

TL;DR

This paper explores the algebraic conditions under which constraint satisfaction problems (CSPs) and SAT problems admit polynomial kernels, focusing on the role of Maltsev and k-edge operations in kernelization.

Contribution

It introduces an algebraic framework linking kernelization limits of SAT and CSP problems to properties like Maltsev and k-edge operations, providing new characterizations.

Findings

CSP problems with Maltsev embeddings have linear kernels.

Embedding into CSPs with k-edge operations yields kernels with O(n^c) constraints.

Maltsev condition may fully characterize SAT problems with linear kernels.

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. 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 is included in co-NP/poly.

Full text

11affiliationtext: Institut für Algebra, TU Dresden, Dresden, Germany22affiliationtext: Department of Computer Science, Royal Holloway, University of London, Great Britain

Kernelization of Constraint Satisfaction Problems: A Study

through Universal Algebra 111This is an extended preprint of Kernelization of Constraint Satisfaction Problems: A Study through Universal Algebra, appearing in Proceedings of the 23rd International Conference on Principles and Practice of Constraint Programming (CP 2017)

Victor Lagerkvist victor.lagerqvist@tu-dresden.de

Magnus Wahlström magnus.wahlstrom@rhul.ac.uk