CSP, Algebras, Varieties

Andrei A. BulatovSimon Fraser University

CSP Reminder

An instance of CSP is defined to be a pair of relational structures A and B over the same vocabulary . Does there exist a homomorphism : A B?

Given a sentence and a model for decide whether or not the sentence is true

iRkn RRxx 11 ,,

Example: Graph Homomorphism, H-Coloring

Example: SAT

CSP(B), CSP(B)

?

G K k

Graph k-Colorability Does there exist a homomorphism from a given graph G to the complete graph?

H-Coloring Does there exist a homomorphism from a given graph G to H?

Example (Graph Colorability)

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 retractionA structure is a core if it has no proper retractionsA structure and its core are homomorphically equivalent it is sufficient to consider cores

Polymorphisms Reminder

Definition A relation (predicate) R is invariant with respect to an n-ary operation f (or f is a polymorphism of R) if, for any tuples the tuple obtained by applying f coordinate-wise is a member of R

Raa n ,,1

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

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

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

From constraint languages to algebras

From algebras to varieties

Dichotomy conjecture, identities and meta-problem

Local structure of algebras

Outline

Languages/polymorphisms vs. structures/algebras

Constraint language},,{ 1 mRR

Set of operations},,{ 1 mffC

Relational structure),,,( 1 mRRA A

Algebra ),,;( 1 mffA A

Inv(C) = Inv(A) Pol( ) = Pol(A)

Str(A) = (A; Inv(A)) Alg(A) = (A; Pol(A ))

CSP(A) = CSP(A)

Algebras - Examples

semilattice operation semilattice (A; )

x x = x, x y = y x, (x y) z = x (y z)

affine operation f(x,y,z) affine algebra (A; f)

f(x,y,z) = x – y + z

group operation group (A; , , 1)-1

permutations kgg ,,1 G-set ),,;( 1 kggA

Expressive power and term operations

),,,( 1 mRRA A ),,;( 1 mffA A

Expressive power),,(,, 11 mm RRRR InvPol

Term operations),,(,, 11 mm ffff Inv Pol

Primitive positive definability

),,,,,(

,,),,(

11

11

nk

nk

yyxx

yyxxR

Substitutions

)),,(,

),,,((),,(

1

111

kn

kk

xxg

xxgfxxh

Clones of relations Clones of operations

Good and Bad

Theorem (Jeavons)

Relational structure is good ifall relations in its expressive power are good

Relational structure is bad ifsome relation in its expressive power is bad

Algebra is good if it hassome good term operation

Algebra is bad if all its term operation are bad

Subalgebras

A set B A is a subalgebra of algebra if every operation of A preserves B

),,;( 1 mffA A

In other words ),,( 1 mffB Inv

It can be made an algebra ),,;( 1 BmB ffB B

}3,2,1,0{ ),;( 44 ZzyxZ {0,2}, {1,3} are subalgebras{0,1} is not

)),,(max;( 4 yxZ any subset is a subalgebra

Subalgebras - Graphs

G = (V,E)

What subalgebras of Alg(G) are?

)),(),(),(( ,)( xzEzyEyxEzyxB

G Alg(G) Inv(Alg(G)) InvPol(G)

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)Take operation f of A and Raaaa nknk ),,(,),,,( 1111

k

nkn

k

bbaa

aaf

,,,,

,,

1

1

111

R

Since is an operation of BBf

Homomorphisms

Algebras and are similar if and have the same arity

),,;( 1A

mA ffA A ),,;( 1

Bm

B ffB BA

ifB

if

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

))(,),(()),,(( 1 nB

innA

i xxfxxf

Homomorphisms - Examples

);( );( 222444 zyxZzyxZ

1 3,10 2,0 :

Affine algebras

Semilattices)};,,({ 1

1 cbaS )};1,0({ 22 S

Homomorphisms - Congruences Let B is a homomorphic image of under homomorphism . Then the kernel of :(a,b) ker() (a) = (b)is a congruence of A

),,;( 1 mffA A

Congruences are equivalence relations from Inv(A)

)(),,(

),,(

1

1 ker

nA

nA

bbf

aaf

)),,(()),,(( 11 nA

nA bbfaaf

))(,),(())(,),(( 11 nB

nB bbfaaf

Homomorphisms - Graphs

)),(),(( ),( yzExzEzyx

G = (V,E)

)),(),(( ),( zyEzxEzyx

congruences of Alg(G)

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() ),,(1 mii xxR

),,)((1

1mii xxR Instance of CSP()

Direct Power

The nth direct power of an algebra is the algebra where the act component-wise

),,;( 1 mffA A),,;( 1

nm

nnn ffA A nif

),,(

),,(,,

11

11111

1

11

nkn

k

nk

k

n

ni

aaf

aaf

a

a

a

af

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

Direct Product - Reduction

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

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

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 x y

(x,y) =

H-Coloring Dichotomy (Cntd)

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

2=

- It has a homomorphic image which is a triangle

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

VarietiesVariety 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

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 x x = x, x y = y x, (x y) z = x (y z)

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

Near-unanimity xyxxfxyxfxxyf ),,,(),,,(),,,(

Constant ),,(),,( 11 kk yyfxxf

Dichotomy Conjecture - Identities

Dichotomy Conjecture A finite algebra A is tractable if and only if var(A) has a Taylor term:

Otherwise it is NP-complete

),,( 1 nzzt

niyxyytxxt iiiiiniini 1 ),,,(),,( 11 for

Idempotent algebrasAn 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

Conservative Algebras

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

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

)()()( xzzyyx zyx

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

semilatticemajorityaffine (Mal’tsev)

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) = yor 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 theyare not necessarily subalgebras

Operations on Divisors

a

b

B

B/

B is the subalgebra generated by a,b

B/ is the quotient algebra modulo

is a maximal congruence of B separating a and b

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

f

Graph of a general algebra

a

b

B

B/

We connect elements a,b with ared edge if there are a maximal congruence of B and a term operation f such that is a semilattice operation on

f},{ ba

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

f},{ ba

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 f

/B

Graph of an Algebra

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

semilatticemajorityaffine

If A is tractable then for any subalgebra B of AGr(B) is connected

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

Dichotomy Conjecture in Colours

The conjecture above is equivalent to the dichotomy conjecture we already have, and it is equivalent to the condition on omitting type 1

Particular cases

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.

babba },{ ba

aabba

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

)(Gr Ar

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!