CSP, Algebras, Varieties Andrei A. Bulatov Simon 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 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 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

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 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 languageSet of operations Relational structure Algebra 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) permutationsG-set

Expressive power and term operations Expressive power Term operations Primitive positive definability Substitutions Clones of relations Clones of operations

Good and Bad Theorem (Jeavons) 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 good if it has some 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 In other words It can be made an algebra {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? 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  R R Since is an operation of B

Homomorphisms Algebras and are similar if and have the same arity A homomorphism of A to B is a mapping  : A  B such that

Homomorphisms - Examples Affine algebras Semilattices

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 Congruences are equivalence relations from Inv (A)   

Homomorphisms - Graphs G = (V,E) 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 (  ) Instance of CSP (  )

Direct Power The n th 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 Product - Reduction Theorem CSP ( ) is poly-time reducible to CSP ( A )

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!!!

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

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 Constant

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

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 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

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 semilattice majority affine (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) = 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

Operations on Divisors a b 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 

Graph of a general algebra a b B B/  We connect elements a,b with a red edge if there are a maximal congruence  of B and a term operation f such that is a semilattice operation on 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 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 an Algebra An edge-colored graph Gr( A ) can be defined on elements of any algebra A semilattice majority affine If A is tractable then for any subalgebra B of A Gr (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. Let denote the graph obtained from Gr( A ) by removing yellow and blue edges

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!