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

Transcription

1 CSP Algebras Varieties Andrei A. Bulatov Simon Fraser University

2 CSP Reminder An instance o CSP is deined to be a pair o relational structures A and B over the same vocabulary τ. Does there exist a homomorphism φ: A B? Example: Graph Homomorphism H-Coloring Given a sentence x1 K xn R1 K R k and a model or R i decide whether or not the sentence is true Example: SAT CSP(B CSP(B

3 Polymorphisms Reminder Deinition A relation (predicate R is invariant with respect to an n-ary operation (or is a polymorphism o R i or any tuples r r a K a R the tuple obtained by applying 1 n coordinate-wise is a member o R Pol(A denotes the set o all polymorphisms o relations o A Pol(Γ denotes the set o all polymorphisms o relations rom Γ Inv(C denotes the set o all relations invariant under operations rom C

4 Outline From constraint languages to algebras From algebras to varieties Dichotomy conjecture identities and meta-problem Datalog and variety Algebras and varieties in other constraint problems

5 Languages/polymorphisms vs. structures/algebras Relational structure A = A R K R ( 1 m Algebra A = ( A ; 1 K m Constraint language Γ = R K R } { 1 m Set o operations C = K } { 1 m Inv(C = Inv(A Str(A = (A; Inv(A Pol(Γ = Pol(A Alg(A = (A; Pol(A CSP(A = CSP(A

6 Algebras - Examples semilattice operation semilattice (A; x x = x x y = y x (x y z = x (y z aine operation (xyz aine algebra (A; group operation (xyz = x y + z -1 group (A; 1 permutations 1 K gk G-set ( A; g1 K gk g

7 Expressive power and term operations ; ( 1 m A K A = ( 1 m R R A K A = Term operations ( 1 1 m m K K Inv = Pol Expressive power ( 1 1 m m R R R R K K InvPol = Substitutions Primitive positive deinability ( ( ( ( k n k k x x g x x g x x h K K K K = ( ( n k n k y y x x y y x x R K K K K Φ = Clones o relations Clones o operations

8 Good and Bad Theorem (Jeavons Relational structure is good i all relations in its expressive power are good Relational structure is bad i some relation in its expressive power is bad Algebra is good i it has some good term operation Algebra is bad i all its term operation are bad

9 Subalgebras A set B A is a subalgebra o algebra i every operation o A preserves B In other words B Inv K ( 1 m A = ( A ; 1 K m It can be made an algebra B = B; K ( 1 B m B ( Z ; x y + z Z4 4 = ( Z4;max( x y {0123} {02} {13} are subalgebras {01} is not any subset is a subalgebra

10 Subalgebras - Graphs G = (VE What subalgebras o Alg(G are? G Alg(G Inv(Alg(G InvPol(G B( x = y z ( E( x y E( y z E( z x

11 Subalgebras - Reduction Theorem (BJeavonsKrokhin Let B be a subalgebra o A. Then CSP(B CSP(A Every relation R Inv(B belongs to Inv(A Take operation o A and ( a11 K a1k K( an1 K a a11 K a1k Since is an operation o B M B an1 K ank b K b R 1 k nk R

12 Homomorphisms A Algebras A = ( A; 1 K m and B = B; A B are similar i and have the same arity i i A B ( 1 K B m A homomorphism o A to B is a mapping ϕ: A B such that A ϕ( i ( xn K xn = i ( ϕ( x1 K ϕ( xn B

13 Homomorphisms - Examples Aine algebras ( Z ; x 4 y + 4 z ( Z2; x 2 y + 2 ϕ : 4 z Semilattices S1 = ({ a b c}; 1 S2 = ({01}; 2

14 Homomorphisms - Congruences Let B is a homomorphic image o A = A ; K under homomorphism ϕ. Then the kernel o ϕ: (ab ker(ϕ ϕ(a = ϕ(b is a congruence o A A A ( a1 K an ( ϕ ( 1 b K b ker n A A 1 n ϕ ( ( a K an = ϕ( ( b1 K b A ( 1 m A 1 n ( ϕ( a K ϕ( an = ( ϕ( b1 K ϕ( b Congruences are equivalence relations rom Inv(A

15 Homomorphisms - Graphs G = (VE ( ( ( ( y z E x z E z y x = θ congruences o Alg(G ( ( ( ( z y E z x E z y x = η

16 Homomorphisms - Reduction Theorem Let B be a homomorphic image o A. Then or every inite Γ Inv(B there is a inite Inv(A such that CSP(Γ is poly-time reducible to CSP( Instance o CSP(Γ R( x i K x i 1 m ϕ 1 ( R( x i K x im 1 Instance o CSP(

17 Direct Power The nth direct power o an algebra is the algebra where the act component-wise ; ( 1 m A K = A ; ( 1 n m n n n A K = A n i = ( ( nk n k nk k n n i a a a a a a a a K M K M K M Observation An n-ary relation rom Inv(A is a subalgebra o n A

18 Direct Product - Reduction Theorem CSP( A n is poly-time reducible to CSP(A

19 Transormations and Complexity Theorem Every subalgebra every homomorphic image and every power o a tractable algebra are tractable Corollary I an algebra has an NP-complete subalgebra or homomorphic image then it is NP-complete itsel

20 H-Coloring Dichotomy Using G-sets we can prove NP-completeness o the k-coloring problem (or -Coloring K k Take a non-bipartite graph H - Replace it with a subalgebra o all nodes in triangles - Take homomorphic image modulo the transitive closure o the ollowing θ(xy = x y

21 H-Coloring Dichotomy (Cntd - What we get has a subalgebra isomorphic to a power o a triangle 2 = - It has a homomorphic image which is a triangle This is a hom. image o an algebra not a graph!!!

22 Varieties Variety is a class o algebras closed under taking subalgebras homomorphic images and direct products Take an algebra A and built a class by including all possible direct powers (ininite as well subalgebras and homomorphic images. We get the variety var(a generated by A Theorem I A is tractable then any inite algebra rom var(a is tractable I var(a contains an NP-complete algebra then A is NP-complete

23 Meta-Problem - Identities Meta-Problem Given a relational structure (algebra decide i 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 Aine Mal tsev (xyy = (yyx = x Near-unanimity ( y x K x = ( x y K x = K = ( x K x y = x Constant ( x K xk = ( y1 K y 1 k

24 Dichotomy Conjecture - Identities Dichotomy Conjecture A inite algebra A is tractable i and only i var(a has a Taylor term: t z K z t Otherwise it is NP-complete ( 1 n K x = t( y K y x y or i 1Kn ( xi1 in i1 in ii ii =

25 Idempotent algebras An algebra is called surjective i every its term operation is surjective I a relational structure A is a core then Alg(A is surjective Theorem It suices to study surjective algebras Theorem It suices to study idempotent algebras: (x x = x I A is idempotent then a G-set belongs to var(a i it is a divisor o A that is a hom. image o a subalgebra

26 Dichotomy Conjecture Dichotomy Conjecture A inite idempotent algebra A is tractable i and only i var(a does not contain a inite G-set. Otherwise it is NP-complete This conjecture is true i - A is a 2-element algebra (Schaeer - A is a 3-element algebra (B. - A is a conservative algebra (B. i.e. every subset o its universe is a subalgebra

27 Complexity o Meta-Problem Theorem (B.Jeavons - The problem given a inite relational structure A decide i Alg(A generates a variety with a G-set is NP-complete - For any k the problem given a inite relational structure A o size at most k decide i Alg(A generates a variety with a G-set is poly-time - The problem given a inite algebra A decide i it generates a variety with a G-set is poly-time

28 Deinability in Datalog Theorem (FederVardi I a relational structure A is such that Pol(A contains a near-unanimity operation then CSP(A is deinable in Datalog I a relational structure A is such that Pol(A contains a semilattice operation then CSP(A is deinable in Datalog

29 Datalog and Algebras Theorem (Larose Zadori Let A A be inite relational structures with the same universe and such that Pol(A Pol(A. Then i CSP(A is deinable in Datalog then so is CSP(A' It makes sense to talk about algebras with CSP(A deinable in Datalog Theorem I CSP(A is deinable in Datalog then so is or any algebra B rom var(a CSP(B

30 Linear Equations Linear Equation: Is a system o linear equations over a inite ield consistent? CSP orm: 2x+ 3y= 5 x+ y 2z= 1 x y z (2x+ 3y= 5 ( x+ y 2z= 1 Linear Equation is equivalent to CSP(A where A = (A; x y + z Fact Linear Equation is not deinable in Datalog

31 Conjecture and Meta-Problem or Datalog Theorem I var(a contains a G-set or an aine algebra then CSP(A is not deinable in Datalog Conjecture CSP(A is deinable in Datalog i var(a does not contain a G-set or an aine algebra The Conjecture is true or 2-3-element and conservative algebras

32 Complexity o Meta-Problem or Datalog Theorem - The problem given a inite relational structure A decide i Alg(A generates a variety with a G-set or an aine algebra is NP-complete - For any k the problem given a inite relational structure A o size at most k decide i Alg(A generates a variety with a G-set or an aine algebra is poly-time - The problem given a inite algebra A decide i it generates a variety with a G-set or an aine algebra is poly-time

33 Localization - Types A G-set aine algebra Possible local structure: 1 G-set 2 linear space 3 2-element Boolean algebra 4 2-element lattice 5 2-element semilattice

34 Omitting Types Conjectures A is tractable i var(a omits type 1 CSP(A is deinable in Datalog i var(a omits types 1 and 2

35 Other Problems: Counting In a #CSP given structures A and B we are asked how many homomorphisms rom A to B are there. Can be parametrized by relational structures in the usual way Can be parametrized by algebras AND varieties For #CSP(A a classiication is also known (B.DalmauGrohe: #CSP(A is solvable in poly-time i var(a has a Mal tsev term (xyy = (yyx = x and every inite algebra rom var(a satisies a certain condition on congruences

36 Other Problems: QCSP In a QCSP we need decide i a positive conjunctive sentence is true in a given model Can be parametrized by relational structures in the usual way Complexity classiication is almost known or 3-element structures (Chen and known or conservative structures (Feder/Chen QCSPs can be parametrized by algebras but not varieties! No easible di/trichotomy conjecture known