Kernelization of Constraint Satisfaction Problems:A Study through Universal Algebra

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

The subpower membership problem of 2-nilpotent algebras

The subpower membership problem SMP(A) of a finite algebraic structure A asks whether a given partial function from A^k to A can be interpolated by a term operation of A, or not. While this problem can be EXPTIME-complete in general, Willard asked whether it is always solvable in polynomial time if A is a Mal'tsev algebras. In particular, this includes many important structures studied in abstract algebra, such as groups, quasigroups, rings, Boolean algebras. In this paper we give an affirmative answer to Willard's question for a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that might be essential in answering the question for general nilpotent Mal'tsev algebras in the future.Comment: 17 pages (including appendix

On the non-efficient PAC learnability of conjunctive queries

This note serves three purposes: (i) we provide a self-contained exposition of the fact that conjunctive queries are not efficiently learnable in the Probably-Approximately-Correct (PAC) model, paying clear attention to the complicating fact that this concept class lacks the polynomial-size fitting property, a property that is tacitly assumed in much of the computational learning theory literature; (ii) we establish a strong negative PAC learnability result that applies to many restricted classes of conjunctive queries (CQs), including acyclic CQs for a wide range of notions of acyclicity; (iii) we show that CQs (and UCQs) are efficiently PAC learnable with membership queries.<p/

The Subpower Membership Problem for Finite Algebras with Cube Terms

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set $\mathcal{K}$ of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP($\mathcal{K}$) is in P if $\mathcal{K}$ is a finite set of finite algebras with a cube term, provided $\mathcal{K}$ is contained in a residually small variety. We also prove that for any finite set of finite algebras $\mathcal{K}$ in a variety with a cube term, each one of the problems SMP($\mathcal{K}$), SMP($\mathbb{HS} \mathcal{K}$), and finding compact representations for subpowers in $\mathcal{K}$, is polynomial time reducible to any of the others, and the first two lie in NP

Sparsification of SAT and CSP Problems via Tractable Extensions

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(nc) 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(n2−ε) for any ε &gt; 0 unless the polynomial hierarchy collapses

Learnability of solutions to conjunctive queries

The problem of learning the solution space of an unknown formula has been studied inmultiple embodiments in computational learning theory. In this article, we study a familyof such learning problems; this family contains, for each relational structure, the problem oflearning the solution space of an unknown conjunctive query evaluated on the structure. Aprogression of results aimed to classify the learnability of each of the problems in this family,and thus far a culmination thereof was a positive learnability result generalizing all previousones. This article completes the classification program towards which this progression ofresults strived, by presenting a negative learnability result that complements the mentionedpositive learnability result. In addition, a further negative learnability result is exhibited,which indicates a dichotomy within the problems to which the first negative result applies.In order to obtain our negative results, we make use of universal-algebraic concepts

Hardness Results for the Subpower Membership Problem

We first provide an example of a finite algebra with a Taylor term whose subpower membership problem is NP-hard. We then prove that for any consistent strong linear Maltsev condition M which does not imply the existence of a cube term, there exists a finite algebra satisfying M whose subpower membership problem is EXPTIME-complete. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show as a corollary that there are finite algebras which generate congruence distributive and congruence k-permutable (k ≥ 3) varieties whose subpower membership problem is EXPTIME-complete. Finally, we show that the spectrum of complexities of the problems SMP() for finite algebras in varieties which are congruence distributive and congruence k-permutable (k ≥ 3) is fuller than P and EXPTIME-complete by giving examples of finite algebras in such a variety whose subpower membership problems are NP-complete and PSPACE-complete, respectively

The Subpower Membership Problem for Finite Algebras with Cube Terms

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set K of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP(K) is in P if K is a finite set of finite algebras with a cube term, provided K is contained in a residually small variety. We also prove that for any finite set of finite algebras K in a variety with a cube term, each one of the problems SMP(K), SMP(HSK), and finding compact representations for subpowers in K, is polynomial time reducible to any of the others, and the first two lie in NP.</p

Learnability of quantified formulas

We consider the following classes of quantified formulas. Fix a set of basic relations called a basis. Take conjunctions of these basic relations applied to variables and constants in arbitrary ways. Finally, quantify existentially or universally some of the variables. We introduce some conditions on the basis that guarantee efficient learnability. Furthermore, we show that with certain restrictions on the basis the classification is complete. We introduce, as an intermediate tool, a link between this class of quantified formulas and some well-studied structures in Universal Algebra called clones. More precisely, we prove that the computational complexity of the learnability of these formulas is completely determined by a simple algebraic property of the basis of relations: their clone of polymorphisms. Finally, we use this technique to give a simpler proof of the already known dichotomy theorem over Boolean domains. © 2003 Elsevier B.V. All rights reserved