Constant query testability of assignments to constraint satisfaction problems

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Chen, Hubie and Valeriote, M. and Yoshida, Y. (2019) Constant-query

testability of assignments to constraint satisfaction problems. SIAM Journal

on Computing 48 (3), pp. 1022-1045. ISSN 1095-7111.

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Constant-Query Testability of Assignments to

Constraint Satisfaction Problems

HUBIE CHEN,

MATT VALERIOTE

_,

YUICHI YOSHIDA

_,

For each finite relational structureA, let CSP(A)denote the CSP instances whose constraint relations are taken fromA. The resulting family of problems CSP(A)has been considered heavily in a variety of computational contexts. In this article, we consider this family from the perspective of property testing: given a CSP instance and query access to an assignment, one wants to decide whether the assignment satisfies the instance, or is far from doing so. While previous work on this scenario studied concrete templates or restricted classes of structures, this article presents a comprehensive classification theorem.

Our main contribution is a dichotomy theorem completely characterizing the finite structures Asuch that CSP(A)is constant-query testable:

• IfAhas a majority polymorphism and a Maltsev polymorphism, then CSP(A)is constant-query testable with one-sided error.

• Else, testing CSP(A)requires a super-constant number of queries.

CCS Concepts: •Theory of computation→Constraint and logic programming;

Sketch-ing and samplSketch-ing;

ACM Reference Format:

Hubie Chen, Matt Valeriote, and Yuichi Yoshida. 2019. Constant-Query Testability of Assign-ments to Constraint Satisfaction Problems. 1, 1 ( January 2019),26pages.https://doi.org/0000001. 0000001

1 INTRODUCTION 1.1 Background

In property testing, the goal is to design algorithms that distinguish objects satisfying

some predetermined propertyP from objects that are far from satisfyingP. More

specifically, forϵ,δ ∈ [0,1], an algorithm is called an(ϵ,δ)-testerfor a propertyP, if

given an inputI, it accepts with probability at least 1−δif the input satisfiesP, and it

∗

Supported by a grant from the Natural Sciences and Engineering Research Council of Canada †

Supported by JST ERATO Grant Number JPMJER1305 and JSPS KAKENHI Grant Number JP17H04676.

Authors’ addresses: Hubie Chen, Birkbeck, University of London, London, United Kingdom, hubie.chen@ ehu.es; Matt Valeriote, McMaster University, Hamilton, Canada, [email protected]; Yuichi Yoshida, National Institute of Informatics, Chiyoda-ku, Tokyo, 101-8430, Japan, [email protected].

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https://doi.org/0000001.0000001

rejects with probability at least 1−δif the inputI isϵ-far from satisfyingP. Roughly

speaking, we say thatIisϵ-farfromPif we must modify more than anϵ-fraction of I to makeI satisfyP. Whenδ =1/3, we simply call it anϵ-tester. A tester is called aone-sided error testerif it always accepts whenIsatisfiesP. In contrast, a standard

tester is sometimes called atwo-sided error tester. As one motivation of property testing

is to design algorithms that run in time sublinear in the input size, we assume query

access to the input, and we measure the efficiency of a tester by itsquery complexity.

We refer to [19,28,29] for surveys on property testing.

Inconstraint satisfaction problems(for short, CSPs), one is given a set of variables and a set of constraints imposed on the variables, and the task is to find an assignment

of the variables that satisfies all of the given constraints. By restricting the relations

used to specify constraints, it is known that certain restricted versions of the CSP

coincide with many fundamental problems such as SAT, graph coloring, and solvability

of systems of linear equations. To formally define these restricted versions of the CSP

(and hence, these problems), we considerrelational structuresA=(A;Γ), whereAis a finite set andΓconsists of a finite set of finitary relations overA. In this context, one often refers toΓas aconstraint languageoverA, and toAas atemplate. Then, we define CSP(A)to be those instances of the CSP whose constraint relations are taken fromΓ. In recent years, computational aspects of CSP(A)have been heavily studied, in the decision setting [2,4,9,22], in counting complexity [10,16], in computational learning theory [14,22], and in optimization and approximation [12,27,31–33]. See also the survey by Barto [3] for an overview of this line of research. Recently, Bulatov [7] and Zhuk [35] have announced proofs of the Feder-Vardi Dichotomy Conjecture, a conjecture that has driven much of the research on the CSP over the past several years. In this paper, we consider the problem family CSP(A) from the perspective of property testing; in particular, we consider the task of testing assignments to CSPs.

Relative to a relational structureA, an input consists of a tuple(I,ϵ,f), whereIis an instance of CSP(A)with weights on the variables,ϵis an error parameter, andf is an assignment toI. In the studied model, the tester has full access toIand query

access tof, that is, a variablexcan be queried to obtain the value off(x). In this sense,

assignment testing lies in themassively parameterized model[26]. We say thatf is ϵ-farfrom satisfyingIif one must modify more than anϵ-fraction off (with respect to the weights) to makef a satisfying assignment ofI, and we say thatf isϵ-close

otherwise. It is always assumed thatIhas a satisfying assignment as otherwise we

can immediately reject the input (in this context, time complexity is not taken into

account). The objective of assignment testing of CSPs is to correctly decide whetherf

is a satisfying assignment ofIor isϵ-far from being so with probability at least 2/3.

Whenf does not satisfyIbut isϵ-close to satisfyingI, the algorithm can output

anything.

In assignment testing, we say that the query complexity of a tester is constant,

sublinear, or linear if it is constant, sublinear, or linear (respectively) in the number of

variables of an instance. The main problem addressed in this paper is to reveal the

1.2 Contributions

While previous work on testing assignments to the problems CSP(A)studied concrete templatesAor restricted classes of structures, this article presents a comprehensive classification of the constant query complexity templates. The results in this paper

were first announced in [15]. Before describing our characterization, we introduce the algebraic notion of apolymorphismwhich is key to the description and obtention of

our results. LetRbe anr-ary relation on a setA. Ak-ary operationf :Ak →Ais said

to be apolymorphismofRif for any length-ksequence of tuples

(a11, . . . , a1

r),(a21, . . . ,ar2), . . . ,(ak1, . . . ,a

k r) ∈R,

implies

(f(a11, . . . , ak

1), . . . ,f(a 1

r, . . . ,akr)) ∈R.

To indicate thatf is a polymorphism ofR, it is also said thatRispreservedbyf. An

operationf is apolymorphismof a relational structureAif it is a polymorphism of each of its relations. We define thealgebraofA, denoted by Alg(A), to be the pair

(A; Pol(A)), where Pol(A)is the set of all polymorphisms ofA.

Definition 1. LetAbe a nonempty set. Amajority operationonAis a ternary operationm : A3 → A such thatm(b,a,a) = m(a,b,a) = m(a,a,b) = a for all a,b ∈ A. AMaltsev operationonAis a ternary operationp : A3 → Asuch that p(b,a,a)=p(a,a,b)=bfor alla,b ∈A.

Theorem 1.1. LetAbe a relational structure. The following dichotomy holds.

(1) IfAhas a majority polymorphism and a Maltsev polymorphism, thenCSP(A)is constant-query testable (with one-sided error).

(2) Else, testingCSP(A)requires a super-constant number of queries.

This theorem generalizes characterizations of the constant-query testableList H-homomorphism problems[34] and Boolean CSPs [6] to general CSPs. In Section3 we will describe the particularly nice structure of relations over templates that have

majority and Maltsev polymorphisms and use this to prove the theorem. For the

moment, let us consider a number of example templates to which this theorem applies.

Example 1. The templateAover the Boolean domain{0,1}whose only relation is_, has both majority and Maltsev polymorphisms. In particular, it is readily verified that

this relation_,is preserved by the Maltsev operation on{0,1}defined byp(x,y,z)= x⊕y⊕z; on the two-element set{0,1}, there is a unique majority operationm, and it is readily verified that_,is preserved bym. Note that CSP(A)coincides with the graph 2-coloring problem.

More generally, templatesAover a finite domain whose relations are graphs of bijections onAhave both majority and Maltsev polymorphisms, since they are

in-stances of the next set of examples (Example2). Instances of CSP(A)for such templates Acoincide with instances of the problem which is the subject of theunique games

conjecture[25]. □

Example 2. Another class of finite structures that have both majority and Maltsev

polymorphisms are those that have adiscriminatoroperation as a polymorphism. On

d(x,y,z)=zand ifx _,y,d(x,y,z)=x. From this definition, it is immediate thatdis a Maltsev operation onA, and thatd(x,d(x,y,z),z)is a majority operation onA. Any

finite product of finite fields will have a discriminator term operation ([11]) and so any finite relational structure whose relations are preserved by the operations of such

a ring will have majority and Maltsev polymorphisms. □

Example 3. Forpa prime number, let_Fpbe the field of sizep, and let_Rbe the ring

F2×F3×F5. Then as noted in Example2,Rhas a discriminator term operation. LetR be the structure with domainRand set of relationsΓconsisting of intersections of the following binary relations onR: Forp=2, 3, or 5,

• C_p ={((a2,a3,a5),(b2,b3,b5)) |ap=bp},

• Fora∈_Fp,D_p,a ={((a2,a3,a5),(b2,b3,b5)) |a_p =a},

• Forb∈_Fp,E_p,b={((a2,a3,a5),(b2,b3,b5)) |b_p =b},

So relations inΓcan express that pairs of elements inRare congruent modulo 2, 3, or 5 in the corresponding coordinate and/or that a certain coordinate is equal to some

fixed value. These relations are invariant under the discriminator term operation of_R

and so according to Theorem1.1, CSP(R)has constant query complexity. □

Examples of structures that satisfy the first condition of Theorem1.1but that do not have a discriminator operation as a polymorphism can be derived from finite Heyting

algebras.

Example 4. Consider the five-element Heyting algebra_Mpresented in [21, Figure 1]. (Heyting algebras are bounded distributive lattices that also have a binary

“impli-cation” operation; they serve as algebraic models of propositional intuitionistic logic.)

This algebra has universeM ={0,a,b,e,1}; the two equivalence relationsαandβ

that partitionMinto blocks{{0,a},{b,e,1}}and{{0,b},{a,e,1}}(respectively) are

preserved by the operations of the algebra. Since_Mhas majority and Maltsev term operations (the operations(x∧y)∨(x∧z)∨(y∧z)and((x→y) →z)∧((z→y) →x)

respectively), then the structureM=(M;α,β)has majority and Maltsev polymor-phisms. The only other non-trivial binary relation onM that is preserved by the

operations of_Misα∩β. □

Example 5. Bulatov and Marx provide yet another example of a structure having

both a majority and a Maltsev polymorphism, in [8, Example 1.1]. Their example is essentially the structure on the 10-element setA = {0,1,2,3,4,5,6,7,8,9}that

has a single ternary relationR={(0,1,2),(0,3,4),(5,6,7),(8,9,7)}. It can be readily

checked that with respect to the usual ordering onA,Ris closed under the ternary

“in-between” operation and so has a majority polymorphism. It can also be checked

thatRhas a ternary polymorphismp(x,y,z)that satisfies the equationsp(x,x,y)= p(x,y,x)=p(y,x,x)=yand so is a Maltsev operation. We note thatRis not preserved by the discriminator operation onA. □

Example 6. Consider the relational structureAover the Boolean domain{0,1} whose only relation is≤. This structure is readily verified to have a majority

poly-morphism (note that over the Boolean domain, there is indeed a unique majority

yields(1,0), which is not in the relation≤. Thus, Theorem1.1implies that CSP(A)is not constant-query testable. From [6] we know that it is sublinear-query testable with

one-sided error. □

To conclude this sub-section we present a theorem that addresses the complexity of

deciding, for a given relational structureA, if CSP(A)is constant-query testable.

Theorem 1.2. The problem of deciding, for a relational structureA, ifCSP(A)is constant-query testable is solvable in polynomial time.

Proof. According to Theorem1.1, deciding if CSP(A)is constant-query testable amounts to deciding ifAhas majority and Maltsev polymorphisms. From [23] it follows that ifBis any structure that has both of these types of polymorphisms then CSP(B) has bounded width. In the terminology of [13], the condition of having majority and Maltsev polymorphisms is a strong linear Maltsev condition. Since it is the case that

a structure will satisfy this condition if and only if the structure obtained from it by

expanding it by all one-element unary relations does then we can apply Lemma 3.8

of [13] to produce a polynomial time algorithm that decides, given a structureA, if it has both majority and Maltsev polymorphisms. □

1.3 Proof outline

We now present an outline of our proof of Theorem1.1.

Ahas majority and Maltsev polymorphisms⇒CSP(A)is constant-query testable. We first look at (1) of Theorem1.1. Let(I,ϵ,f)be an input to assignment testing of CSP(A). First, we preprocessIso that it becomes 2-consistent and reject ifIhas no solution (see Section3for the formal definition). Using the 2-consistency ofIand the majority polymorphism ofAwe can assume that for each variablex ofI, the set of allowed values forxforms a domainA_xthat is the universe of an algebra_Ax

that is a factor (i.e., a homomorphic image of a subalgebra) of Alg(A), the algebra of polymorphisms ofA. Also, we can assume that for each pair of variablesx,yofI there is a unique binary constraint ofIwith scope(x,y)and constraint relationR_xy,

withR_xy the universe of some subalgebra of_Ax×_Ay. Furthermore these are the only

constraints ofI.

In order to test whetherf satisfiesI, we use three types of reductions: a factoring

reduction, a splitting reduction, and an isomorphism reduction. Each reduction pro-duces an instanceI′and an assignmentf′such thatf′satisfiesI′iff satisfiesI,

andf′isΩ(ϵ)-far from satisfyingI′iff isϵ-far from satisfyingI. For simplicity, we focus on how we create a new instanceI′here.

The objective of the factoring reduction is to factor, for each variablexofI, the

domainA_x by any congruenceθ of_Ax (i.e., an equivalence relation onA_x that is

preserved by the operations of_Ax) for which none of the constraint relations ofI

distinguish betweenθ-related values ofA_x.

After ensuring that all of the domainsA_xofIcannot be factored, we then employ a

splitting reduction to ensure that for each variablexofIthe algebra_Axis subdirectly

irreducible, i.e., cannot be represented as a subdirect product of non-trivial algebras.

For any variablexfor which_Axcan be represented as a subdirect product of

and the domainA_x by the domainsA1_x andA2_x. For any other variableyofI, we

“split” the constraint relationR_yx (and its inverseR_xy) into two relationsR_yx

1andRyx2 that are together equivalent to the original one. We then add these two new relations

(and their inverses) toI, along withA_x, now regarded as a binary relation from the

variablex1tox2.

After performing the splitting reduction and the factoring reduction, we next define a

binary relation∼on the set of variables ofIsuch thatx∼yif and only if the constraint

relationR_xy is the graph of an isomorphism from_Ax to_Ay. Using 2-consistency and

the fact that the domains ofIare subdirectly irreducible and cannot be factored, it follows that, unlessIis trivial, the relation∼will be a non-trivial equivalence relation.

Within each∼-class, the domains are isomorphic via the corresponding constraint

relations ofI, and this allows us to produce an isomorphism-reduced instanceI′by

restrictingIto a set of variables representing each of the∼-classes.

After performing this isomorphism reduction, the resulting instance may have

domains which can be further factored, allowing us to apply the factoring reduction

to produce a smaller instance. We show that if we reach a point at which none of

the three reductions can be applied, the instance must be trivial, either having just a

single variable, or for which|A_x|=1 for all variablesx. We also show that this point

will be reached after applying the reductions at most|A|-times.

In Section3, we will see how these reductions work on the template in Example3.

CSP(A)is constant-query testable⇒Ahas majority and Maltsev polymorphisms. Now we look at (2) of Theorem1.1. We show that ifAdoes not have these two types of polymorphisms, then we cannot test CSP(A)with a constant number of queries. We use the fact that having these two types of polymorphisms is equivalent

toAhaving a Maltsev polymorphism and that the variety of algebras generated by Alg(A)iscongruence meet semidistributive[20]. When the variety generated by Alg(A) is not congruence meet semidistributive, then it can be easily shown from [6,34] that testing CSP(A)requires a linear number of queries. WhenAdoes not have a Maltsev polymorphism, we show that there exists a structureA′having a binary non-rectangular relation such that we can reduce CSP(A′)to CSP(A). Then, by replacing the 2-SAT relations with this binary non-rectangular relation, we can reuse the argument

for showing a super-constant lower bound for 2-SAT in [17] to obtain a super-constant lower bound for CSP(A).

1.4 Related work

Assignment testing of CSPs was implicitly initiated by [17]. There, it was shown that 2-CSPs are testable withO(

√

n)queries and requireΩ(logn/log logn)queries for any fixedϵ >0. On the other hand, 3-SAT [5], 3-LIN [5], and Horn SAT [6] requireΩ(n) queries to test.

is a tuple(G,{L_v}_v∈V(G),ϵ,f), whereGis a (weighted) graph,L_v ⊆V(H) (v∈V(G))

are list constraints,f :V(G) →V(H)is a mapping given as a query access, andϵis an

error parameter. The goal is testing whetherf is a listH-homomorphism fromGor ϵ-far from being so, whereϵ-farness is defined analogously to testing assignments of CSPs. It was shown in [34] that the algebra (or the variety) associated with the listH -homomorphism characterizes the query complexity, and that listH-homomorphism is

constant-query (resp., sublinear-query) testable if and only ifH is a reflexive complete

graph or an irreflexive complete bipartite graph (resp., a bi-arc graph).

Testing assignments of Boolean CSPs was studied in [6], and in that paper relational structures were classified into three categories: (i) structuresA for which CSP(A) is constant-query testable, (ii) structuresAfor which CSP(A)is not constant-query testable but sublinear-query testable, and (iii) structuresAfor which CSP(A)is not sublinear-query testable. They also relied on the fact that algebras (or varieties) can

be used to characterize query complexity.

1.5 Organization

Section2introduces the basic notions used throughout this paper. We show the constant-query testability of CSPs with majority and Maltsev polymorphisms in

Section3. Super-constant lower bounds of CSPs without majority or Maltsev poly-morphisms is discussed in Section4.

2 PRELIMINARIES

For an integerk, let[k]denote the set{1, . . . ,k}.

Constraint satisfaction problems.For an integerk ≥1, ak-ary relation on a domain Ais a subset ofAk. Aconstraint languageon a domainAis a finite set of relations on A. A(finite) relational structure, or simply a(finite) structureA=(A;Γ)consists of a (finite) nonempty setAand a constraint languageΓonA.

For the remainder of this paper we will assume that all relational structures that

are mentioned are finite. For a structureA =(A;Γ), we define the problem CSP(A) as follows. An instanceI = (V,A,C,w)consists of a set of variablesV, a set of constraintsC, and a non-negative weight functionwwithÍ_x∈Vw(x)=1. Here, each constraintC ∈ Cis of the form⟨(x1, . . . ,x_k),R⟩, wherex1, . . . ,x_k ∈V are variables, Ris a relation inΓandkis the arity ofR. The tuple(x1, . . . ,x_k)is called thescopeof the constraintCandRis called theconstraint relationofC. AnassignmentforIis a

mappingf :V →A, and we say thatf is asatisfying assignmentiff satisfies all the

constraints, that is,(f(x1), . . . ,f(xk)) ∈Rfor every constraint⟨(x1, . . . ,xk),R⟩ ∈ C.

Algebras and Varieties:Let_A=(A;F)be an algebra. A setB ⊆Ais asubuniverseof

Aif every operationf ∈Frestricted toBhas image contained inB. For a nonempty subuniverseBof an algebra_A,f|_Bis the restriction off toB. The algebra_B=(B,F|_B),

whereF|_B ={f|_B | f ∈F}is asubalgebraof_A. Algebras_A,_Bare of thesame typeif

they have the same number of operations and corresponding operations have the same

arities. Given algebras_A,_Bof the same type, theproduct_A×_Bis the algebra with the

same type as_Aand_Bwith universeA×Band operations computed coordinate-wise.

A subalgebra_Cof_A×_Bis asubdirect productof_Aand_Bif the projections ofCtoA

An equivalence relationθ onAis called acongruenceof an algebra_Aifθ is the

universe of a subalgebra of_A×_A. The collection of congruences of an algebra naturally

forms a lattice under the inclusion ordering, and this lattice is called thecongruence latticeof the algebra. Given a congruenceθ of _A, we can form the homomorphic imageA/θ, whose elements are the equivalence classes ofθand the operations are defined so that the natural mapping from_Ato_A/θis a homomorphism. An operation f(x1, . . .xn)on a setAisidempotentiff(a,a, . . . ,a)=afor alla∈A, an algebra_Ais idempotentif each of its operations is, and a class of algebras is idempotent if each of its members is. We note that if_Ais idempotent, then for any congruenceθof_A, the θ-classes are all subuniverses of_A.

Avariety is a class of algebras of the same type closed under the formation of

homomorphic images, subalgebras, and products. For any algebra_A, there is a smallest

variety containing_A, denoted byV(_A)and called thevariety generated by_A. It is

well known that any variety is generated by an algebra and that any member ofV(_A)

is a homomorphic image of a subalgebra of a power of_A.

Many important properties of the algebras in a variety can be correlated with

properties of the congruence lattices of it member algebras. In this work we consider

several congruence lattice conditions for varieties, including congruence distributivity,

congruence meet semidistributivity, and congruence permutability. Details of these

conditions can be found in [20] and more details on the basics of algebras and varieties can be found in [11].

2.1 Assignment problems

An assignment problemconsists of a set ofinstances, where each instance I has

associated with it a set of variablesV, a domainA_vfor each variablev ∈V, and a weight functionw :V → [0,1]withÍ_v∈Vw(v) =1. An assignment ofIis a mappingf defined onVwithf(x) ∈A_xfor each variablex ∈V. Each instanceIof an assignment

problem has associated with it a notion of asatisfying assignment. For two assignments f andдforI, we define their distance as dist_I(f,д):=Í_x∈V:f(x)

,д(x)w(x). We define dist_I(f)=min_дdist_I(f,д), whereдis over all satisfying assignments ofI. Then, for ϵ ∈ [0,1], we say that an assignmentf forIisϵ-far from satisfyingIif dist_I(f)>ϵ. In theassignment testing problemcorresponding to an assignment problem, we are

given an instanceIof the assignment problem,ϵ ∈ [0,1], and a query access to

an assignmentf forI, that is, we can obtain the value of f(x)by queryingx ∈V.

Then, we say that an algorithm is atester for the assignment problem if it accepts

with probability at least 2/3 whenf is a satisfying assignment ofI, and rejects with

probability at least 2/3 whenf isϵ-far from satisfyingI. Thequery complexityof a

tester is the number of queries tof.

We can naturally view CSP(A)as an assignment problem: for each instance on a set of variablesV, the associated assignments are the mappings fromV toA, and the notion of satisfying assignments is as described above. Note that an input to the

assignment testing problem corresponding to CSP(A)is a tuple(I,ϵ,f), whereIis an instance of CSP(A),ϵis an error parameter, andf is an assignment toI. In order to distinguishIfrom the tuple(I,ϵ,f), we always call the formerinstanceand the

2.1.1 Gap-preserving local reductions.We will frequently use the following reduc-tion when constructing algorithms as well as showing lower bounds.

Definition 2 (Gap-preserving local reduction). Given assignment problemsP andP′, there is a(randomized) gap-preserving local reduction fromPtoP′if there exist a functiont(n)and constantsc1,c2satisfying the following: given aP-instance Iof with variable setV and an assignmentf forI, there exist aP′-instanceI′with

variable setV′_{and an assignmentf}′_forI′_{such that the following hold:} (1) |V′| ≤t(|V|).

(2) Iff is a satisfying assignment ofI, thenf′is a satisfying assignment ofI′.

(3) For anyϵ ∈ (0,1), ifdist_I(f) ≥ϵ, thenPr[dist_I′(f′) ≥c1ϵ] ≥9/10holds, where the probability is over internal randomness.

(4) Any query tof′can be answered by making at mostc2queries tof.

Alinear reductionis defined to be a gap-preserving local reduction for which the

functiont(n)isO(n).

Lemma 2.1 ([34]). LetPandP′be assignment problems. Suppose that there exists an ϵ-tester forP′with query complexityq(n,ϵ)for anyϵ ∈ (0,1), wherenis the number of variables in the given instance ofP′, and that there exists a gap-preserving local reduction

fromPtoP′with functiont. Then, there exists anϵ-tester forPwith query complexity

O(q(t(n),O(ϵ)))for anyϵ >0, wherenis the number of variables in the given instance of

P. In particular, linear reductions preserve constant-query and sublinear-query testability.

As another application of gap-preserving local reductions, the following fact is

known.

Lemma 2.2 (Lemma 6.4 and 6.5 of [34]). LetA,A′be relational structures. If the relations ofAare preserved by the operations of some finite algebra inV(Alg(A′)), and CSP(A′)is constant-query testable, thenCSP(A)is constant-query testable.

3 CONSTANT-QUERY TESTABILITY

In this section, assume thatA=(A;Γ)is a structure that has a majority polymorphism m(x,y,z)and a Maltsev polymorphismp(x,y,z). It is known [11] that this is equiva-lent to the varietyAgenerated by the algebra Alg(A)beingcongruence distributive andcongruence permutable. This means that for each algebra_B∈ A, the lattice of

congruences of_Bsatisfies the distributive law; and, that for each pair of congruences αandβof_B, the relationsα◦βandβ◦αare equal. Such varieties are also said to be arithmetic.

An important feature ofA (and in fact of any congruence distributive variety

generated by a finite algebra) is that every subdirectly irreducible member ofAhas

size bounded by|A|([11]). We will make use of the fact that an algebra issubdirectly irreducibleif and only if the intersection of all of its trivial congruences is non-trivial. This is equivalent to the algebra having a smallest non-trivial congruence.

In this section, we will show that CSP(A)is constant-query testable. Some of the ideas found in this section were inspired by the paper [8].

For our analysis, it is useful to introduce the problem CSP(V), for each variety

⟨(x1, . . . ,x_k),R⟩, whereR is the domain of a subalgebra _Rof _Ax

1 × · · · ×Axk. In particular,Ris also the domain of an algebra inV. The definition of an assignment

testing problem naturally carries over to instances of CSP(V).

LetI=(V,{A_x}_x∈V,C,w)be an instance of CSP(A). SinceAhas a majority term, we can assume that each constraint inCis binary [1]. Furthermore, we may assume thatIhas a solution and is 2-consistent:

• for everyx,y ∈V, there is a unique constraintC_xy =⟨(x,y),R_xy⟩ofIwith

scope(x,y)and the constraint relationR_xy is a subdirect product ofA_x andA_y,

• forx ∈V,R_xx is the equality relation 0_A

x on the setAx, and

• ifx,y,z ∈ V and (a,b) ∈ R_xy then there is an elementc ∈ A_z such that

(a,c) ∈Rxzand(b,c) ∈R_yz.

Note that from 2-consistency it follows that for allx,y ∈V,R_yx =R−1_xy ={(b,a) |

(a,b) ∈Rxy}for anyx,y∈V. Under these assumptions, we may writeIas

(V,{Ax}x∈V,{Rxy}(x,y)∈V2,w)

or simplyI = (V,{A_x},{R_xy},w). It is well known that any CSP instance over a template having a majority polymorphism can be transformed to a 2-consistent

instance of the form just described in polynomial time without changing the set of

satisfying assignments; see [23] or [8] for more details. So, there is no loss in generality in assuming throughout the rest of this section that any instance of CSP(A)considered

will be 2-consistent and have only binary constraints.

SinceAis assumed to be congruence permutable, then for anyx _,y ∈V, the binary relationR_xy isrectangular, that is,(a,c),(a,d),(b,d) ∈R_xy implies(b,c) ∈R_xy

(in Lemma4.8we show the converse, i.e., a failure of congruence permutability implies a failure of rectangularity). As noted in Lemma 2.10 of [8], this is equivalent toR_xy being athick mapping. This means that there are congruencesθ_xy of_Axandθ_yx of_Ay

such that modulo the congruenceθ_xy×θ_yx on_Rxy, the relationR_xy is the graph of

an isomorphismϕ_xy from_Ax/θ_xy to_Ay/θ_yx and such that for alla∈A_xandb ∈A_y,

(a,b) ∈R_xy if and only ifϕ_xy(a/θ_xy)=b/θ_yx. In this situation, we say thatR_xy is a

thick mapping with respect toθ_xy,θ_yx andϕ_xy. For future reference, we note that if

for some variablesx _,y, the congruenceθ_xy =0_A

x then the relationRyx is the graph of a surjective homomorphism from_Ayto_Ax.

3.1 A factoring reduction

LetI=(V,{A_x},{R_xy},w)be a 2-consistent instance of CSP(A)and for eachx ∈V letµ_x =Ó_y

,xθxy, a congruence ofAx. We say thatAx isprimeifµxis the equality congruence 0_A

x andfactorableotherwise. Roughly speaking, ifAxis not prime, then we can factorA_xbyµ_x without changing the problem, because no constraint ofI

distinguishes values within anyµ_x-class. Formally, we define the factoring reduction

as in Algorithm1.

Let(I,ϵ,f)be an input of CSP(A)and let(I′,ϵ′,f′)=Factor(I,ϵ,f). It is clear

that since the instanceIof CSP(A)is assumed to be 2-consistent then the instance

I′_{will also be 2-consistent. Furthermore, the sizes of the domains ofI}′_{are no larger}

than the sizes of the domains ofI. Now we show that the factoring reduction is a

Algorithm 1

1: procedureFactor(I=(V,{A_x},{R_xy},w),ϵ,f) 2: forx∈V do

3: Ax ←Ax/µx.

4: f(x) ←f(x)/µx.

5: for(x,y) ∈V ×V do

6: R_xy ← {(a/µ_x,b/µ_y) | (a,b) ∈R_xy}.

7: return(I,ϵ,f).

Lemma 3.1. Let(I,ϵ,f)be an input ofCSP(A)and let(I′,ϵ′,f′)=Factor(I,ϵ,f). If(I′_,ϵ′_,f′_{)is testable withq(ϵ}′_{)queries, then(I,ϵ,f)is testable withq(O(ϵ))queries.}

Proof. We show that the factoring reduction is a linear reduction. Let the original and reduced instances be

I=(V,{Ax},{Rxy},w) and I′=(V′,{A_x′}_x∈V,{R_xy′ },w′)

respectively.

Note that|V′| = |V|and we can determine the value of f′(x)by querying f(x)

once.

Iff satisfiesI, thenf′also satisfiesI′. Suppose thatf′isϵ-close to satisfyingI′

and letд′be a satisfying assignment ofI′with dist_I′(f′,д′) ≤ ϵ. Then, we define дto be any assignment forIsuch that forx ∈V, iff′(x)=д′(x)thenд(x)= f(x) and iff′(x)_,д′(x), thenд(x)is taken to be an arbitrary element in theµ_x-classд′(x).

Then,дsatisfiesIand dist_I(f,д)=dist_I′(f′,д′) ≤ϵ.

To summarize, the factoring reduction is a gap-preserving local reduction with t(n)=n,c1=1, andc2=1. □

Example 7. [Example3, continued] Let(I,ϵ,f)be an input of CSP(R), where

I=(V,{A_x},{R_xy},w)

is a 2-consistent instance. So eachA_xis equal toA2×A3×A5where eachApis eitherFp

or{a}for somea∈F_p. For anyx ∈V,µ_xwill be a congruence onA_xand will be equal

to the kernel of a projection map onto some of the factors ofA_x. So, after applying

Factor, the resulting instance will have domains that are isomorphic to a product

of one, two, or three of the setsF2,F3, andF5, with the corresponding constraints

reduced accordingly. □

3.2 Reduction to instances with subdirectly irreducible domains

In this section, we provide a reduction that produces instances whose domains are all

subdirectly irreducible. Suppose that_Ais a subdirect product of two algebras_A1,A2

fromAand that_Ris a subdirect product of_Aand_Bfor some_B∈ A. We can project

the relationRonto the factors of_Ato obtain two new binary relations fromA1toB

and fromA2toB, respectively:

The following shows that the relationRcan be recovered from the relationsR1,R2,

andA(considered as a relation fromA1toA2).

Lemma 3.2. For alla1∈A1,a2 ∈A2, andb∈B, the following are equivalent:

• ((a1,a2),b) ∈R

• (a1,b) ∈R1,(a2,b) ∈R2and(a1,a2) ∈A.

Proof. One direction of this claim follows by construction. For the other, suppose

that(a1,b) ∈R1,(a2,b) ∈R2and(a1,a2) ∈A. Then there are elementsci ∈Ai, for i =1, 2, with(a1,c2),(c1,a2) ∈ A,((a1,c2),b),((c1,a2),b) ∈ R. SinceR is subdirect inA×Band(a1,a2) ∈Athen there is somed ∈ Bwith((a1,a2),d) ∈R. Applying

the majority term of A coordinate-wise to the tuples((a1,c2),b),((c1,a2),b), and

((a1,a2),d)fromRwe produce the tuple((a1,a2),b) ∈R, as required. □

Lemma3.2allows us to split a domain of an instance of CSP(A)into subdirectly irreducible domains. Formally, we define the splitting reduction as in Algorithm2.

Algorithm 2

1: procedureSplit(I=(V,{A_x},{R_xy},w),ϵ,f)

2: whilethere existsx ∈V such that_Ax is not subdirectly irreducible or trivial

do

3: Replace _Ax inI with an isomorphic non-trivial subdirect product of

Ax1×Ax2for some quotientsAx1,Ax2ofAxsuch thatAx1is subdirectly irreducible.

4: V ← (V\ {x}) ∪ {x1,x2}, wherex1andx2are newly introduced variables. 5: Remove the domainA_xand add the domainsA_x

1andAx2over the variables x1andx2respectively.

6: C ← C \ {⟨(x,x),R_xx⟩,⟨(x,y),R_xy⟩,⟨(y,x),R_yx⟩}_y∈V\{x}.

7: C ← C ∪ {⟨(x1,x1),0_Ax

1⟩,⟨(x2,x2), 0_Ax

2⟩,⟨(x1,x2),Ax⟩,⟨(x2,x1),A −1

x ⟩}.

8: C ← C∪{⟨(x1,y),(Rxy)

1⟩,⟨(x2,y),(Rxy)2⟩,⟨(y,x1),(Rxy) −1

1 ⟩,⟨(y,x2),(Rxy) −1

2 ⟩}y∈V\{x}. 9: Removexfrom the domain ofw and addx1andx2.

10: Setw(x1)=w(x)/2 andw(x2)=w(x)/2.

11: Removexfrom the domain off and addx1andx2.

12: Setf(x1) ∈Ax1andf(x2) ∈Ax2so that(f(x1),f(x2))=f(x).

13: return(I,ϵ/2|A|,f).

Let(I,ϵ,f)be an input of CSP(A)and let(I′,ϵ′,f′)=Split(I,ϵ,f). It is clear

that, sinceIis assumed to be a 2-consistent instance of CSP(A)then the splitting

reduction constructs another 2-consistent instanceI′of CSP(A)whose domains are

all subdirectly irreducible and so have size bounded by|A|(and are no bigger than the

domains ofI). The next lemma shows that splitting domains of an instance does not

affect the primeness of the instance’s domains.

Lemma 3.3. LetI′be the instance ofCSP(A)obtained by splitting a domain_Axof another instanceIinto two subdirect factorsAx1andAx2as in the Split procedure. If the domainAx is prime inIthen the domainsAx1andAx2are prime inI

′_{. If}

Ayis

Proof. LetI =(V,{A_x},{R_xy},w)be given and suppose that the domain_Ax is a subdirect product of the algebras_Ax

1andAx2. To produceI

′_{fromIby splitting}

Ax, we replace the variablexand the domainA_x with the variablesx1andx2and the corresponding domainsA_x

1andAx2. For eachy∈V withx,y, we replace the constraint⟨(x,y),R_xy⟩with the constraints⟨(x1,y),(R_xy)

1⟩and⟨(x2,y),(Rxy)2⟩and add the constraint⟨(x1,x2),A_x⟩.

If the domain_Axis prime inIthen there isk ≥1 and variablesy_i ∈V\ {x}, for

1≤i≤k, such thatÓ_1≤i≤kθ_xyi =0_Ax. To show that_Ax

1is prime inI ′

it will suffice

to show that

Û

1≤i≤k θx1yi

!

∧θ_x1x2=0Ax1.

To establish this, suppose that(a1,a′

1)belongs to the left hand side of this equality. We will show thata1=a

′

1. We have that(a1,a ′

1) ∈θx1yi for 1≤i ≤kand(a1,a ′ 1) ∈ θx1x2. From the latter membership it follows that there is somec ∈ Ax2 such that

(a1,c),(a1,c′ ) ∈A_x. From(a1,a′

1) ∈θx1yi it follows that there is someu ∈Ayi with

(a1,u),(a′

1,u) ∈ (Rxyi)1. We can conclude that there ared,d ′_∈A

yi with((a1,d),u),

((a′ 1,d

′_{),u) ∈ R}

xyi. We then have that((a1,d),(a1′,d ′_{)) ∈ θ}

xyi. We can now apply the majority term ofA coordinate-wise to the following three pairs of members

ofθ_xy

i to establish that((a1,c),(a ′

1,c)) ∈θxyi:((a1,d),(a ′ 1,d

′₎₎

,((a1,c),(a1,c)), and

((a1′,c),(a ′

1,c)). We’ve shown that(a1,c)and(a ′

1,c)areθxyi-related for alli ≤kand so we have that(a1,c)=(a′

1,c), which implies thata1=a ′

1, as required. ThusAx1is prime inI′and by symmetry,_Ax

2is also prime.

A similar use of the majority polymorphism can establish the last part of this

lemma. □

Now we show that the splitting reduction is a gap-preserving local reduction.

Lemma 3.4. Let(I,ϵ,f)be an input ofCSP(A)and let(I′,ϵ′,f′)=Split(I,ϵ,f). If(I′,ϵ′,f′)is testable withq(ϵ′)queries, then(I,ϵ,f)is testable withq(O(ϵ))queries.

Proof. We show that the splitting reduction is a linear reduction.

LetI=(V,{A_x},{R_xy},w)andI′=(V′,{A′_x},{R′_xy},w′)be the original instance and the reduced instance, respectively.

In the reduction, every variablexofV is ultimately split into variablesx1, . . . ,x_k x fromV′and the domain_Axis replaced by subdirectly irreducible domains_A1_x, . . . ,_Ak_xx

corresponding to these variables such that_Axis isomorphic to a subdirect product of

these new domains. Since each of the domains has size bounded by|A|, thenk_x ≤ |A|

for allx ∈V and so after completely splitting_Ax into thek_x factors, we have that w(x) ≤2|A|w′(x_i)for eachi ∈ [k_x]. We also have thatÍ_i∈[k

x]w′(xi)=w(x)for each x∈V.

We can determine the value off′(x_i), wherex_i is added when splitting the variable x, we only need to know the value off(x).

Iff satisfiesI, thenf′satisfiesI′by Lemma3.2. Suppose thatf′isϵ/(2|A|)-close to satisfyingI′and letд′be a satisfying assignment forI′with dist_I′(f′,д′) ≤ϵ/(2|A|).

Because the tuple(д′(x1), . . . ,д ′_(x

kx))is inAx, we can naturally define an assignment дforIby settingд(x)=(д′(x1), . . . ,д

′_(x

from Lemma3.2. Moreover,

dist_I(f,д)=

Õ

x∈V:∃i∈[k_x],д′(x_i)_,f′(x_i) w(x)

≤Õ

x∈V

Õ

i∈[kx]:д′_(x

i)_,f′_(x

i)

2|A|w′(x_i)

=2|A|dist_I′(f′,д′) ≤ϵ.

To summarize, the splitting reduction is a gap-preserving local reduction with t(n)=|A|,c1=1/2|A|, andc2=1. □

Example 8. [Example3, continued] After applying the procedure Split each of the domains of the resulting instance will be trivial or isomorphic toF_pfor somep =2,

3, or 5. For variablesx _,y, the binary constraint fromA_xtoA_y will either be trivial

(i.e., equal toA_x×A_y) or equal to the graph of an isomorphism fromA_xtoA_y. Since

this new instance will be reduced, then for any non-trivialA_ythere will be at least

onex, with the latter holding. □

3.3 Isomorphism reduction

By applying the factoring reduction and then the splitting reduction to an instance of

CSP(A)we end up with an instance whose domains are either trivial or subdirectly

irreducible and prime. For such an instance, we have the following property.

Lemma 3.5. LetI=(V,{A_x},{R_xy},w)be an instance ofCSP(A)such that|V| >1 and such that every domain is either trivial or is subdirectly irreducible and prime. Then, for each variablex ∈V, there is at least one variabley_,xso thatθxy =0_A

x and for such variablesy, the relationRyx is the graph of a surjective homomorphism fromAyto

Ax.

Proof. If|A_x| =1 then the result follows trivially. Otherwise, we have that the

congruenceµ_x =Ó_y

,xθxy ofAx is equal to 0Ax, sinceAx is prime. But, since this algebra is subdirectly irreducible, it follows that for somey _,x,θ_xy =0_A

x. Since Ryx is a thick mapping withθ_xy =0_Ax it follows thatR_yx is the graph of a surjective

homomorphism from_Ayto_Ax. □

LetI=(V,{A_x},{R_xy},w)be an instance of CSP(A)with|V| >1 and with the property that every domain is either trivial or is subdirectly irreducible and prime.

Define the relation∼onV byx ∼yif and only if the relationR_xyis the graph of an

isomorphism from_Axto_Ay. Using the 2-consistency ofI, the relation∼is naturally

an equivalence relation onV. The following corollary to Lemma3.5establishes that unless all of the domains ofIare trivial, the relation∼is non-trivial.

Corollary 3.6. ForI=(V,{A_x},{R_xy},w)an instance ofCSP(A)as in Lemma3.5, ifx∈V is such that the domainAxhas maximal size and has at least two elements, then

there is somey∈V withx ,yandx ∼y.

Proof. IfA_x has maximal size and has at least two elements, then lety∈V be a

variable such thatx _,yandR_yx the graph of a surjective homomorphism from_Ayto

Ax. SinceA_x has maximal size, it follows that|A_y|=|A_x|and soR_yx is the graph of

For a variablex ∈V, let[x]:=x/∼denote the∼-class ofV thatxbelongs to. Let S⊆V be an arbitrary complete system of representatives of this equivalence relation and for any∼-classu, lets(u) ∈V be the unique elementx ∈Ssuch thatx ∈u. In

particular[s(u)]=uholds.

Given an assignmentf forI, we can test the input(I,ϵ,f)in two steps. First, we

test whether the values off in the∼-classes ofV are consistent using a consistency

algorithm (Algorithm3) and then we test the input obtained by contracting the∼ -classes using Algorithm4. Explanations of these two steps are contained in the next two subsections.

3.3.1 Testing∼-consistency. We say that the input(I,ϵ,f)is∼-consistentif, for eachx,y∈V withx ∼y,(f(x),f(y)) ∈R_xy.

For a∼-classu⊆V andb∈A_s(u), we define

w(u,b)= Õ

y∈u:f(y)=R_s(u)y(b) w(y),

w(u)= Õ

b∈As(u)

w(u,b), and

wmaj(u)= max

b∈A_s(u)

w(u,b).

Note thatw(u)is also equal toÍ_x∈uw(x), the sum of the weights of the variables inu. In addition, we defineϵ_uto be(w(u)−wmaj(u))/w(u)and observe thatϵu ≤ (|A|−1)/|A| since|A_s(x)| ≤ |A|and sow(u)is the sum of at most|A|terms, each of which is at mostwmaj(u). The quantityϵu represents the fraction of values, by weight, of f|u that need to be altered in order to establish∼-consistency of the assignment over the

classu. Let fmajbe the assignment obtained fromf in this way. That is, forx ∈V, fmaj(x)=Rs([x])x

argmax_b∈A

s([x])w([x],b)

.

We need the following simple proposition to analyze our algorithm.

Proposition 3.7. LetXbe a random variable taking values in[0,1]such that_E[X] ≥ ϵ for someϵ ≥0. Then,Pr[X ≥ϵ/2] ≥ϵ/2holds.

Proof. Letp=Pr[X ≥ϵ/2]. Then,

ϵ ≤E[X] ≤1·p+ϵ 2

(1−p) ≤p+ϵ 2 .

Hence,p ≥ϵ/2 holds. □

In order to test∼-consistency, we run Algorithm3.

Lemma 3.8. Algorithm3tests∼-consistency with query complexityO(1/ϵ2).

Proof. It is clear that Algorithm3accepts iff is∼-consistent and the query com-plexity isO(1/ϵ2). Suppose that f isϵ-far from∼-consistency, which means that

dist_I(f,fmaj) ≥ϵ. Then, we haveE[ϵu]= Í

u:∼-class

w(u)ϵ_u ≥ϵ, where in the calculation

of the expectation, a∼-classuis chosen with probabilityw(u). Note thatϵ_u ∈ [0,1] for every∼-classuand so we can apply Lemma3.7, to conclude that we sample a

∼-classuwithϵ_u ≥ϵ/2 with probability at leastϵ/2. Hence, the probability thatU

Algorithm 3

1: procedureConsistency(I,ϵ,f)

2: Sample a setU ofΘ(1/ϵ) ∼-classes ofI. In each sampling,uis chosen with probabilityw(u).

3: foreachu∈U do

4: LetSbe the set obtained by samplingΘ(1/ϵ)variables inuwith replace-ment. In each sampling, a variablex ∈uis chosen with probabilityw(x)/w(u). 5: ifthere are two variablesx,y∈Swithf(y)_,R_xy(f(x))then

6: Reject.

7: Accept.

Algorithm 4

1: procedureIsomorphism(I,ϵ,f)

2: foreach∼-classudo

3: Sample a variablex ∈u with probabilityw(x)/w(u), and letx_u be the sampled variable.

4: V′←V′∪ {u}.

5: A′_u ←As₍u₎.

6: w′(u) ←w(u). 7: f′(u) ←Rx

us(u)(f(xu)).

8: foreach pair(u,u′)of∼-classesdo

9: R′

uu′←Rx_ux_u′.

10: return((V′,{A′_x},{R_xy′ },w′),ϵ/2,f′).

hidden constant large enough. For a∼-classuwithϵ_u ≥ϵ/2, the probability that we

find two verticesx,y∈uwithf(y)_,R_xy(f(x))inSis at least

1− (1−ϵ_u)Θ(1/ϵ)− (ϵ_u)Θ(1/ϵ)≥1− (1−ϵ/2)Θ(1/ϵ)− ((|A| −1)/|A|)Θ(1/ϵ) (1)

sinceϵ_u ≥ϵ/2 for this classuand, as noted earlier,ϵ_u ≤ (|A| −1)/|A|for every class u. By choosing the hidden constant large enough we can ensure that (1) is at least 5/6. By combining these bounds, we obtain two verticesx,ywithf(y)_,R_xy(f(x))with

probability at least 2/3. □

3.3.2 Isomorphism reduction.Using Algorithm3, we can reject an input(I,ϵ,f)if it is far from satisfying∼-consistency. In this subsection we will consider a reduction

from(I,ϵ,f)to another input(I′,ϵ′,f′)assuming that it has not been rejected by

Algorithm3.

Our reduction, as described in Algorithm4, contracts the variables in each∼-class to a single variable from that class. It should be clear that since the instanceI of

CSP(A)is assumed to be 2-consistent, the reduction will produce another 2-consistent

instanceI′of CSP(A). As the next lemma shows, unless the domains ofIall have

size one, some of the domains ofI′will no longer be prime.

If some domain ofIhas more than one element, then any domain ofI′_{of maximal size} will not be prime, unlessI′_{has only one variable.}

Proof. Suppose thatI′has more than one variable. This is equivalent to there

being more than one∼-class forI. Letx be a variable ofI′with|A_x|of maximal

size and letybe any other variable ofI′. Note that according to the construction of

I′fromI, bothxandyare also variables ofIwithx _≁y. Furthermore, |A_x|has

maximal size amongst all of the domains ofIand so the relationR_yx cannot be the

graph of a surjective homomorphism from_Ayto_Ax. If it were, then it would be the

graph of an isomorphism, contradicting thatx _≁y. Thus the congruenceθ_{xy ,}0_Ax. Since_Ax is subdirectly irreducible it follows thatµ_x =Ó_y

,xθxy is also not equal to 0_A

x and soAxis not prime inI′. □

Lemma 3.10. Let(I,ϵ,f)be an input ofCSP(A)and suppose thatf isϵ/20-close to satisfying∼-consistency. Let(I′,ϵ′,f′)=Isomorphism(I,ϵ,f). If(I′,ϵ′,f′)is testable

withq(ϵ′)queries, then(I,ϵ,f)is testable withq(O(ϵ))queries.

Proof. We show that the reduction in Algorithm4is a linear reduction. LetI=

(V,{A_x},{R_xy},w)andI′ =(V′,{A′_x},{R_xy′ },w′)be the original instance and the

reduced instance, respectively.

Note that|V′| ≤ |V|and we can determine the value of f′(u)by queryingf(x_u).

Also, if f satisfiesI, then it is clear thatf′satisfiesI′.

We want to show that, iff is far from satisfyingI, thenf′is also far from satisfying

I′_{with high probability. To this end, we first show that the following quantity is}

small with high probability:

dist(f,f′):= Õ

u:∼-class

Õ

x∈u:

f′_(u)

,Rx s(u)(f(x))

w(x).

For a∼-classu, we define

dist_u(f,f′):=1−w

(u,f′_(u)) w(u) =

Õ

x∈u:

f′_(u)

,Rx s(u)(f(x))

w(x)

w(u).

Note that we have dist(f,f′)= Í

u:∼-class

w(u)dist_u(f,f′).

Then for any∼-classu,

E

xu

[dist_u(f,f′)]= Õ

b∈A_s(u)

w(u,b)

w(u)

1−w

(u,b)

w(u)

≤wmaj(u)

w(u)

1−w maj(u) w(u)

+

1−w

maj(u) w(u)

·1

≤2

1− wmaj(u)

w(u)

=2ϵ_u.

Thus,E_{x

u}u:∼-class[dist(f,f

′_{)]is equal to}

E {x_u}

h Õ

u:∼-class

w(u)dist_u(f,f′) i

≤ Õ

u:∼-class

2w(u)ϵ_u ≤ ϵ 10

.

From Markov’s inequality, we have Pr_{x

Algorithm 5

1: procedureIsomorphism′(I,ϵ,f)

2: ifConsistency(I,ϵ/20,f) rejectsthen

3: Reject.

4: else

5: returnIsomorphism(I,ϵ,f)

Letд′be a satisfying assignment forI′closest tof′. We define an assignmentдfor

Iasд(x)=R_s([x])x(д′([x])). It is clear thatдis a satisfying assignment. Since we have

dist(f,f′)+dist(f′,д′) ≥dist(f,д) ≥ϵ, it follows that Pr[dist(f′,д′) ≥ϵ/2] ≥19/20.

To summarize, the isomorphism reduction is a gap-preserving local reduction with t(n) ≤n,c1=1/2, andc2=1. □

Example 9. [Example3, continued] The∼-classes of the current version of our instance will consist of domains that are pairwise isomorphic to each other via the

corresponding constraint relations. After performing the Isomorphism reduction on

this instance, we will end up with an instance whose constraint relations are trivial,

i.e., for variablesx_,y,R_xy =A_x×A_y. After further reducing this instance via the

Factor reduction, we will end up with an instance whose domains all have size equal

to one. □

Finally, we combine Algorithm3and Algorithm4. to produce Algorithm5and make use of it in the following.

Lemma 3.11. Let(I,ϵ,f)be an input ofCSP(A)and suppose that Isomorphism′(I,ϵ,f) returned another instance(I′,ϵ′,f′). If(I′,ϵ′,f′)is testable withq(ϵ′)queries, then (I,ϵ,f)is testable withq(O(ϵ))queries.

Proof. Consider Algorithm5. Iff satisfiesI, then the∼-consistency test always accepts, and hence we always accept with probability 2/3 from Lemma3.10. Suppose thatf isϵ-far from satisfyingI. Iff isϵ/20-far from satisfying∼-consistency, then the

∼-consistency test rejects with probability at least 2/3. Iff isϵ/20-close to satisfying

∼-consistency, then we reject with probability at least 2/3 by Lemma3.10. □

3.4 Putting things together

Combining the reductions introduced so far we can design a shrinking reduction,

which shrinks the maximum size of the domains of an instance of CSP(A).

Lemma 3.12. Let(I,ϵ,f)be an input ofCSP(A), and suppose that Shrink(I,ϵ,f) returned another instance(I′_,ϵ′_,f′_{). If we can test(I}′_,ϵ′_,f′_)withq(ϵ′_{)queries, then} we can test(I,ϵ,f)withq(O(ϵ))queries. Moreover, the reduction reduces the maximum

size of a domain of the given input, if this maximum is greater than one and the reduced instance has more than one variable.

Proof. We note that at each step of the algorithm, the domains of the instances

that are produced are no larger than the domains of the original instance. Furthermore,

if any of the domains of the original instance has size greater than one, then it follows

Algorithm 6

1: procedureShrink(I,ϵ,f)

2: (I,ϵ,f) ←Factor(I,ϵ,f).

3: (I,ϵ,f) ←Split(I,ϵ,f).

4: ifIsomorphism′(I,ϵ,f) rejectsthen

5: Reject.

6: else

7: (I,ϵ,f) ←the input returned by Isomorphism′.

8: (I,ϵ,f) ←Factor(I,ϵ,f).

9: return(I,ϵ,f).

smaller than that of the original instance, as long as the output instance has more

than one variable. □

Theorem 3.13. LetAbe a structure that has majority and Maltsev polymorphisms. Then,CSP(A)is constant-query testable with one-sided error.

Proof. By applying the shrinking reduction at most|A|times, we get an instance

for which every variable has a domain of size one or which has only one variable. In

either case, the testing becomes trivial. □

4 NON CONSTANT-QUERY TESTABILITY

In this section we consider structuresAthat do not have a majority polymorphism or do not have a Maltsev polymorphism. As noted in the previous section, this is the

same as the varietyV(Alg(A))failing to be arithmetic. For such structures we will show that CSP(A)is not constant-query testable:

Theorem 4.1. If the relational structureAdoes not have a majority polymorphism or does not have a Maltsev polymorphism, thenCSP(A)is not constant-query testable.

Proof. From [20] we know that for a structureA, having both majority and Maltsev polymorphisms is equivalent toV(Alg(A))being congruence meet semidistributive and congruence permutable. The former and latter cases are handled by Theorems4.7 (Section4.1) and4.11(Section4.2), respectively. □

4.1 Hardness for the non congruence meet semidistributive case

Suppose that V(Alg(A)) is not congruence meet semidistributive. We define the singleton-expansionof Ato beA′ = (A,Γ∪ {{a} | a ∈ A}). We first observe that CSP(A′)will be sublinear-query testable if CSP(A)is. Although this observation for the Boolean case was already given in Lemma 5 of [6], its proof was not published yet, and hence we provide the proof for the general case here for completeness.

Lemma 4.2. LetA′be the singleton-expansion ofA. Assume thatϵ ≪ 1

2|A|. IfCSP(A) is testable withq(n,ϵ)queries, thenCSP(A′)is testable withq(O(n),O(ϵ))+Θ(1/ϵ) queries.

Proof. Suppose we can test CSP(A)withq(n,ϵ)queries. Given an instanceI′=

(V′_,A,C′_,w′_{)of CSP(A}′_{),ϵ ≪} 1

2|A|, and a query access to an assignmentf

we want to test whether f′is a satisfying assignment or isϵ-far from being so. For a ∈ A, defineV_a ⊆ V′to be the set of all variablesv for which there is a unary constraint((v),{a})inC′. We assume

Õ

{w(v) |a∈A,v∈V_a,f′(v)_,a} ≤ϵ (2)

as otherwise we can reject f′with high probability by samplingΘ(1/ϵ)variables uniformly at random.

Now, we define a set of variablesV =(V′\Ð_a∈AV_a) ∪ {x_a}_a∈Aand define a set

of constraintsCby removing fromC′all unary constraints and by identifying all

variables inV_awith a new variablex_afor eacha∈A. Next, we definew :V → [0,1] byw(v)=w′(v)/(1+2ϵ|A|)for eachv∈V′\Ð_a∈AV_aandw(x_a)=2ϵfor eacha∈A. LetI=(V,A,C,w)be an instance of CSP(A).

Now given an assignmentf′:V′→Ato the variables ofI′, define an assignment f :V →Ato the variables ofIby setting f(v)=f′(v)for eachv ∈V′\Ð_a∈AV_a andf(x_a) =afor eacha ∈ A. Clearly, iff′satisfiesI′, then f satisfiesI. On the

other hand, supposef isϵ-close to a satisfying assignmentf˜forI. Then, we must

have f˜(x_a)=afor everya ∈ Afrom our choice ofw(x_a). Define f˜′ :V′ → Aby setting f˜′(v)= f˜(v)for everyv ∈V′\Ð_a∈AV_aand f˜′(v)=afor eacha ∈Aand v ∈V_a. Then,f˜′satisfiesI′. From the assumption (2), the distance betweenf′and

˜

f′_{is at mostϵ(1+2ϵ|A|)+ϵ ≤3ϵ. Thus, we have a gap-preserving local reduction} from CSP(A′)to CSP(A), and so, Lemma2.1finishes the proof. □

By adding all of the unary singleton relations toAto produceA′it follows that the varietyV(Alg(A′))is idempotent and will also not be congruence meet semidis-tributive. This is because whether or not an algebra generates a congruence meet

semidistributive variety depends solely on its idempotent term operations (see

Theo-rem 8.1 of [24]). For such a structure, the following is known:

Lemma 4.3. LetA′be the structure as above. Then, there is some finite algebra_Bin

V(Alg(A′))and some subuniverseγof_B3whose domain can be identified with_Fℓ

pk for some primepand integersk, ℓ ≥1such thatγ ={a+b+c=0|a,b,c∈_Fℓ

pk}.

Proof. A combination of Theorem 4.3 and Proposition 2.1 from [18] implies that there is some finite algebra_BinV(Alg(A′))(in fact_Bwill be isomorphic to a quotient of a subalgebra of Alg(A′)) that is either term equivalent to the algebra with universe

{0,1}having no basic operations or is term equivalent to the idempotent reduct

of a module over a finite ring. Theorem 2.1 of [30] provides more detail on this module: it can be taken to be the module_Fℓ

pk over the ring ofℓ×ℓmatrices over the finite field_Fpk for some prime numberpand some integersk, ℓ ≥1. In this case, γ ={a+b+c=0|a,b,c∈_Fℓ

pk}is a subuniverse ofB 3

.

In the first case where_Bis term equivalent to the algebra with universe{0,1}

having no basic operations,γ ={a+b+c =0|a,b,c∈_F2}is a subuniverse ofB 3

since every subset ofB3will be a subuniverse. □

To this end, we introduce some definitions. For a vertex setSin a graph, letN1(S)

be the set of itsunique neighbors, that is, vertices with exactly one neighbor inS.

Forλ,γ >0, we say that a bipartite graph(L,R;E)is a(λ,γ)-right unique neighbor expander if|N1(S)| > λ|S|holds for anyS ⊆ R with|S| ≤γn. For a CSP instance

I=(V,A,C,w), we define itsprimal graphG(I)as the bipartite graph(V,C;E)such

that the pair(v,C) ∈V × Cbelongs toEif and only ifvis in the scope ofC. Now, we

show the following:

Lemma 4.4. Letp be a prime andB =(_Fp;Γ)be a constraint language such thatΓ contains a relation{(a,b,c) | a+b+c =0}. Then, testingCSP(B)requires a linear number of queries even when the primal graph of the input instance is restricted to be a

(λ,γ)-right unique neighbor expander for some universal constantλandγ.

Proof. As the proof is almost identical to that for the casep=2 in [5], we only highlight the difference.

Showing the hardness for the casep=2 amounts to finding a subspace ofU ⊆_Fn 2 such that the basis{a1, . . . ,a_m}of the dual spaceU⊥satisfies the following properties:

• The basis isϵ-separating, that is, everyx ∈_Fn

2 with a uniquei ∈ [m]satisfying

⟨ai,x⟩ ,0 has|x| ≥ϵn.

• The basis is(q,µ)-local, that is, everyα ∈_Fn

2 that is a sum of at leastµmvectors in the basis has|α| ≥q,

Here,ϵ > 0 is the error parameter, and we want to chooseq =Ω(n)andµto be a small constant, say, 1/10. We can reuse this argument for our case by changing the

Hamming weights|x|and|α|with theℓ0norms∥x∥0and∥α∥0, that is, the numbers non-zero elements in those vectors.

Ben-Sassonet al.[5] constructed such a subspace from a random regular bipartite graph. More specifically, given a bipartite graphG=(L,R;E), they constructed a CSP

instance on the variable setLwith a constraint of the formÍ_u∈N(v)u=0 (mod 2)

for each right vertexv ∈R, whereN(v) ⊆Lis the set of neighbors ofv. We use the

same construction by regardinguas an element in_Fpinstead of_F2.

For a vertex setS, letNp(S)be the set of neighbors ofSwhose neighbors inSis

non-zero modulop. Forλ,γ > 0, we say that a bipartite graph(L,R;E)is a(λ,γ) -rightp-expanderif|Np(S)|>λ|S|for anyS⊆Rwith|S| ≥γn. Ben-Sassonet al.[5] showed that the obtained subspace isϵ-separating and(q,µ)-local by using the fact

that a random regular bipartite graph(L,R;E)is a(λ,γ)-right 2-expander with high

probability for some suitableλandγ. Similarly, we can show that a random regular

bipartite graph is a(λ,γ)-rightp-expander with high probability and that the obtained

subspace isϵ-separating and(q,µ)-local. □

Next, we generalize Lemma4.4to the case thatk ≥1.

Lemma 4.5. Letpbe a prime,k ≥1be an integer, andB=(_Fpk;Γ)be a constraint language such thatΓcontains a relation{(a,b,c) |a+b+c=0}. Then, testingCSP(B) requires a linear number of queries even when the primal graph of the input instance is restricted to be a(λ,γ)-right unique neighbor expander for some universal constantλ

andγ.

λ,γ >0 by Lemma4.4. We show a gap-preserving local reduction from CSP(B′)to CSP(B).

Given an instanceI′=(V′,_Fp,C′,w′)of CSP(B′)such that the primal graphG(I′) is a(λ,γ)-right unique neighbor expander, we construct an instanceI=(V,_Fpk,C,w), whereV =V′,C=C′(after changing the domain from_Fpto_Fpk), andw =w′. A value

in_Fpk can be identified with a vector in_Fk_p, where addition in_Fpk is coordinate-wise

addition in_Fk_p. Now given an assignmentf′:V →_Fpto the variables ofI′, define an

assignmentf :V →_Fpk to the variables ofIby settingf(v)=(f′(v),0, . . . ,0) ∈_Fk_p

for everyv ∈ V. Clearly, if f′satisfiesI′, then f satisfiesI. On the other hand,

suppose f isϵ-close to a satisfying assignment f˜ forI. LetS = {v ∈ V | ∃i >

1 withf˜(v)(i)_,0}. Then, if a constraint of the formx+y+z=0 involves a variable

inS, then we must have another variable inSin the constraint. This violates the fact

thatG(I′)is(λ,γ)-right unique neighbor expander (in the regimeϵ <γ), and hence S=∅holds. Then, we can naturally recover a satisfying assignmentf˜′forI′fromf˜ by settingf˜′(v)=f˜(v)(1), which isϵ-close to f′. □

We further generalize to the case thatℓ≥1. We omit the proof because it is almost

identical to that of Lemma4.5.

Lemma 4.6. Letpbe a prime,k, ℓ ≥1be integers, andB=(_Fℓ

pk;Γ)be a constraint language such thatΓcontains a relation{(a,b,c) |a+b+c=0}. Then, testingCSP(B) requires a linear number of queries.

Theorem 4.7. LetAbe a relational structure such thatV(Alg(A))is not congruence meet semidistributive. Then, testingCSP(A)requires a linear number of queries.

Proof. Immediate from Lemmas2.2,4.2,4.3, and4.6. □

4.2 Hardness for the non congruence permutable case

Now, we consider the case thatV(Alg(A))is not congruence permutable. We use the following well-known fact.

Lemma 4.8. LetAbe a finite relational structure that does not have a Maltsev poly-morphism. Then, there is some finite algebraBinV(Alg(A))and some subuniverseγ of

B2such that there are elements0,1∈Bwith(0,0),(0,1),(1,1) ∈γ and(1,0)_<γ.

Proof. SinceA does not have a Maltsev polymorphism, thenV(Alg(A))is not congruence permutable and so there is some finite algebra_B ∈ V(Alg(A))having congruencesαandβsuch thatα◦β_,β◦α. We may assume thatα◦β _⊈β◦αand

so there will be elements 0, 1∈Bwith(0,1) ∈α◦βbut(1,0)_<α◦β. Sinceα◦βis a

reflexive relation, then settingγ =α◦βworks. □

We now establish a super-constant lower bound for CSP((B;γ))forBandγ as in

Lemma4.8based on the super-constant lower bound for monotonicity testing given in [17]. We first note that it is not clear whether we can directly reduce monotonicity testing to testing CSP((B;γ))to obtain a super-constant lower bound for the latter

problem. The reason is thatBmay have more than two elements andγ may have