CSP, Algebras, Varieties

1. CSP, Algebras, Varieties Andrei A. Bulatov Simon Fraser University

2. 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)

3. 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?

4. 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

5. 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

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

7. 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)

8. -1 group (A; , , 1) permutations G-set 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 

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

10. 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

11. 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

12. Subalgebras - Graphs G = (V,E) What subalgebras of Alg(G) are?  Alg(G) G  Inv(Alg(G))  InvPol(G)

13. 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

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

15. Homomorphisms - Examples Affine algebras Semilattices

16. 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)

17. Homomorphisms - Graphs G = (V,E) congruences of Alg(G)

18. 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()

19. 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

20. Theorem CSP( ) is poly-time reducible to CSP(A) Direct Product - Reduction

21. 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

22. (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

23. 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!!!

24. 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

25. 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 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 Constant

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. Thank you!