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CSP, Algebras, Varieties

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CSP, Algebras, Varieties

Andrei A. Bulatov

Simon Fraser University

Given a sentence

and a model for decide whether or not the

sentence is true

CSP Reminder

An instance of CSPis defined to be a pair of relational

structures A and B over the same vocabulary .

Does there exist a homomorphism : A B?

Example: Graph Homomorphism, H-Coloring

Example: SAT

CSP(B), CSP(B)

G

K

k

Example (Graph Colorability)

Graph k-Colorability Does there exist a homomorphism

from a given graph G to the complete graph?

?

H-ColoringDoes there exist a homomorphism from a given

graph G to H?

Cores

A retraction of a relational structure A is a homomorphism onto its substructure B such that it maps each element of B to itself

A core of a structure is a minimal image of a retraction

A structure is a core if it has no proper retractions

A structure and its core are homomorphically equivalent

it is sufficient to consider cores

Definition

A relation (predicate) R is invariantwith respect to ann-ary

operation f (or f is a polymorphismof R) if, for any tuples

the tuple obtained by applying f

coordinate-wise is a member of R

Polymorphisms Reminder

Pol(A) denotes the set of all polymorphisms of relations of A

Pol() denotes the set of all polymorphisms of relations from

Inv(C) denotes the set of all relations invariant under

operations from C

Outline

- From constraint languages to algebras
- From algebras to varieties
- Dichotomy conjecture, identities and meta-problem
- Local structure of algebras

Algebra

Relational structure

Constraint language

Set of operations

Languages/polymorphisms vs. structures/algebras

Inv(C) = Inv(A)

Pol() = Pol(A)

Str(A) = (A; Inv(A))

Alg(A) = (A; Pol(A))

CSP(A)=CSP(A)

-1

group (A; , , 1)

permutations

G-set

Algebras - Examples

semilattice operation

semilattice (A; )

xx = x, xy = yx, (xy) z = x(yz)

affine operation f(x,y,z)

affine algebra (A; f)

f(x,y,z) = x – y + z

group operation

Term operations

Expressive power

Expressive power and term operations

Substitutions

Primitive positive definability

Clones of relations

Clones of operations

Good and Bad

Theorem (Jeavons)

Algebra is good if it has

some good term operation

Relational structure is good if

all relations in its expressive

power are good

Relational structure is bad if

some relation in its expressive

power is bad

Algebra is bad if all its term

operation are bad

A set B A is a subalgebra of algebra

if every operation of A preserves B

In other words

It can be made an algebra

Subalgebras

{0,2}, {1,3} are subalgebras

{0,1} is not

any subset is a subalgebra

Subalgebras - Graphs

G = (V,E)

What subalgebras of Alg(G) are?

Alg(G)

G

Inv(Alg(G))

InvPol(G)

Take operation f of A and

Since is an operation of B

Subalgebras - Reduction

Theorem (B,Jeavons,Krokhin)

Let B be a subalgebra of A. Then CSP(B) CSP(A)

Every relation R Inv(B) belongs to Inv(A)

R

Algebras and

are similar if and have the same arity

A homomorphism of A to B is a mapping : A B

such that

Homomorphisms

Homomorphisms - Examples

Affine algebras

Semilattices

Let B is a homomorphic image of

under homomorphism . Then the kernel of :

(a,b) ker() (a) = (b)

is a congruence of A

Homomorphisms - Congruences

Congruences are equivalence relations from Inv(A)

Homomorphisms - Graphs

G = (V,E)

congruences of Alg(G)

Instance of CSP()

Homomorphisms - Reduction

Theorem

Let B be a homomorphic image of A. Then for every

finite Inv(B) there is a finite Inv(A) such that

CSP() is poly-time reducible to CSP()

Instance of CSP()

The nth direct power of an algebra

is the algebra where the

act component-wise

Observation

An n-ary relation from Inv(A) is a subalgebra of

Direct Power

Theorem

CSP( ) is poly-time reducible to CSP(A)

Direct Product - Reduction

Transformations and Complexity

Theorem

Every subalgebra, every homomorphic image and every

power of a tractable algebra are tractable

Corollary

If an algebra has an NP-complete subalgebra

or homomorphic image then it is NP-complete itself

(x,y) =

x

y

H-Coloring Dichotomy

Using G-sets we can prove NP-completeness of the

H-Coloring problem

Take a non-bipartite graph H

- Replace it with a subalgebra of all nodes in triangles

- Take homomorphic image
- modulo the transitive
- closure of the following

2

H-Coloring Dichotomy (Cntd)

- What we get has a subalgebra isomorphic to a power of
- a triangle

=

- It has a homomorphic image which is a triangle

This is a hom. image of an algebra, not a graph!!!

Varieties

Variety is a class of algebras closed under taking subalgebras,

homomorphic images and direct products

Take an algebra A and built a class by including all possible direct powers (infinite as well), subalgebras, and homomorphic images.

We get the variety var(A) generated by A

Theorem

If A is tractable then any finite algebra from var(A) is

tractable

If var(A) contains an NP-complete algebra then A is

NP-complete

Near-unanimity

Meta-Problem - Identities

Meta-Problem

Given a relational structure (algebra), decide if it is tractable

HSP Theorem A variety can be characterized by identities

Semilattice xx = x, xy = yx, (xy) z = x(yz)

Affine Mal’tsev f(x,y,y) = f(y,y,x) = x

Constant

Dichotomy Conjecture

A finite algebra A is tractable if and only if

var(A) has a Taylor term:

Otherwise it is NP-complete

Dichotomy Conjecture - Identities

Idempotent algebras

An algebra is called surjective if every its term operation is surjective

If a relational structure A is a core then Alg(A) is surjective

Theorem

It suffices to study surjective algebras

Theorem

It suffices to study idempotent algebras: f(x,…,x) = x

If A is idempotent then a G-set belongs to var(A) iff it is a divisor of A, that is a hom. image of a subalgebra

Dichotomy Conjecture

Dichotomy Conjecture

A finite idempotent algebra A is tractable if and

only if var(A) does not contain a finite G-set.

Otherwise it is NP-complete

This conjecture is true if

- A is a 2-element algebra (Schaefer)

- A is a 3-element algebra (B.)

- A is a conservative algebra, (B.)
- i.e. every subset of its universe is a subalgebra

Complexity of Meta-Problem

Theorem (B.,Jeavons)

- The problem, given a finite relational structure A,
- decide if Alg(A) generates a variety with a G-set,
- is NP-complete

- For any k, the problem, given a finite relational
- structure A of size at most k, decide if Alg(A)
- generates a variety with a G-set, is poly-time

- The problem, given a finite algebra A, decide if it
- generates a variety with a G-set, is poly-time

Theorem (B.)

A conservative algebra A is tractable if and only if, for any

2-element subset C A, there is a term operation f

such that f is one of the following operations:

semilattice operation (that is conjunction or disjunction)

majority operation ( that is )

affine operation (that is )

C

Conservative Algebras

An algebra is said to be conservative if every its subset is a

subalgebra

semilattice

majority

affine (Mal’tsev)

Graph of an Algebra

If algebra A is conservative and is tractable, then an

edge-colored graph Gr(A) can be defined on elements of A

GMM operations

An (n-ary) operation f on a set A is called a generalized

majority-minority operation if for any a,b A operation f on

{a,b} satisfies either `Mal’tsev’ identities

f(y,x,x,…,x) = f(x,…,x,y) = y

or the near-unanimity identities

f(y,x,x,…,x) = f(x,y,x,…,x) =…= f(x,…,x,y) = x

Theorem (Dalmau)

If an algebra has a GMM term then it is tractable

We again have two types of pairs of elements, but this time they

are not necessarily subalgebras

B/ is the quotient algebra modulo

B/

For an operation f of A the corresponding operation on B/ is denoted by

Operations on Divisors

B is the subalgebra generated by a,b

B

- is a maximal congruence of B
separating a and b

b

a

B

red edge if there are a maximal congruence of

B and a term operation f such that is a

semilattice operation on

b

yellow edge if they are not connected with a red

edge and there are a maximal congruence of

B and a term operation f such that is a

majority operation on

a

B/

blue edge if they are not connected with a red or

yellow edge and there are a maximal

congruence of B and a term operation f

such that is an affine operation on

Graph of a general algebra

We connect elements a,b with a

Graph of an Algebra

An edge-colored graph Gr(A) can be defined on elements of

any algebra A

semilattice

If A is tractable then for

any subalgebra B of A

Gr(B) is connected

majority

affine

Dichotomy Conjecture in Colours

Dichotomy Conjecture

A finite idempotent algebra A is tractable if and

only if for any its subalgebra B graph Gr(B) is

connected.

Otherwise it is NP-complete

The conjecture above is equivalent to the dichotomy conjecture we already have,

and it is equivalent to the condition on omitting type 1

Observe that red edges are directed, because a semilattice operation on can act as

and the this edge is directed from a to b. Or it can act as

and in this case the edge is directed from b to a.

Let denote the graph obtained from Gr(A) by removing yellow and blue edges

Particular cases

Particular cases II

Theorem (B.)

If for any subalgebra B of

algebra A graph

is connected and has a

unique maximal strongly

connected component, then

A is tractable

- commutative conservative grouppoids a b = b a,
- a b {a,b}

- 2-semilattices a a = a, a b = b a,
- a (a b) = (a a) b

Thank you!