CSP: Algorithms and Dichotomy Conjecture

1. CSP: Algorithms and Dichotomy Conjecture Andrei A. Bulatov Simon Fraser University

2. Definition: Instance: (V;A;C) where  V is a finite set of variables  A is a set of values  C is a set of constraints Question: whether there is h: V  A such that, for any i, is true where each belongs to  Constraint Satisfaction Problem I CSP()

3. Constraint Satisfaction Problem II Q(u,v,w) u - R(w,x) v - R(x,y) S(y,u) w - x - y -

4. u v w x Examples: 3-COL 3-COL =CSP()

5. 3-SAT = CSP( ): Examples: Linear Equations, SAT Linear Equations:

6. Definition A relation R is invariant with respect to ann-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 Invariants and Polymorphisms Pol() denotes the set of all polymorphisms of relations from 

7. If are solutions then Polymorphisms: Affine Relations Affine relations: Relations that can be represented by a system of linear equations Let also (affine operation)

8. Polymorphisms: 2-SAT 2-clauses give rise to binary relations Let (median operation) Operation h is a polymorphism of

9. Polymorphisms: 3-COL  does not have any polymorphisms except for very trivial ones, e.g. f(x,y,z)=y

10. Theorem (Jeavons; 1998) If  ,  are constraint languages such that Pol( )  Pol( ), then CSP( ) is log space reducible to CSP( ) 1 1 2 2 2 1 Larose, Tesson, 2007: This reduction can be made Polymorphisms and Complexity

11. 2 1 1 1 2 3 5 0 0 6 4 Good Polymorphisms: Semilattice A semilattice operation is a binary operation  satisfying the equations: x x = x, x  y = y  x, x  (y  z) = (x  y)  z A semilattice operation induces a partial order: a  b  a  b = b gcd(x,y) max(x,y)  There is always a unique maximal element

12. u - v - w - x - y - Good Polymorphisms: Semilattice

13. u - v - w - x - y - Good Polymorphisms: Semilattice Propagation

14. Chinese Remainder Theorem for Majority Let R be a (k-ary) relation invariant under a majority operation, and is some tuple. Then if for any i,j {1,...,k} there is a tuple such that then Good Polymorphisms: Majority A majority operation is a ternary operation h that satisfies the equations h(x,x,y) = h(x,y,x) = h(y,x,x) = x

15. u - v - w - x - y - Good Polymorphisms: Majority Propagation again: 2-consistency Any 2-consistent instance has a solution

16. Good Polymorphisms: Affine An affine operation is a ternary operation m that is given by x – y + z where +, – are operations of a certain Abelian group CSP over a language invariant under an affine operation is just solving systems of linear equations Gaussian Elimination

17. Complexity: Boolean CSP CSP() is Boolean if  is over {0,1} Theorem (Schaefer 1978) For a constraint language  over {0,1} the problem CSP() is solvable in poly time iff  has a semilattice, majority, or affine polymorphism; otherwise it is NP-complete Fine Print:`Trivial’ languages are excluded from the theorem. These are so-called 0- or 1-valid languages, in which every instance has a solution

18. Complexity: Graphs If  consists of a single binary relation (thought of being the edge relation of some (di)graph H), then CSP() is also called H-Coloring Theorem (Hell, Nesetril 1990) For a graph H the H-Coloring problem is solvable in poly time iff E(H) has a majority polymorphism; otherwise it is NP-complete Fine Print:Graphs here must be cores. Then a core has a majority polymorphism iff it is a loop or an edge

19. Types 5 types of local structure. Defined by the presence of polymorphisms that locally act as one of the 3 good polymorphisms • unary none • affine only affine • boolean all three • lattice majority, semilattice • semilattice semilattice Fine Print: One needs to be quite creative to relate this definition to the actual definition as it was introduced in universal algebra 25 years ago. It is good enough for our purpose, though

20. Conjectures  omits a type if it does not exhibit local structure of this type, otherwise it admits it • (Dichotomy Conjecture) CSP() is solvable in polynomial time iff  omits the unary type; NP-complete otherwise • CSP() is in NL iff  omits the unary, affine, and semilattice types • CSP() is in L iff  omits the unary, affine, lattice and semilattice types • CSP() is in Mod L iff  omits the unary, lattice, and semilattice types  p

21. Algorithms B., Jeavons, Krokhin 2006: if  admits the unary type, CSP() is NP-complete Jeavons et al. mid 90s: algorithms if  has one of the 3 good polymorphisms Barto, Kozik 2010; B. 2010: propagation works iff  omits the unary and affine type Idziak, et al. 2010: the Generalized Gaussian elimination algorithm if  omits the unary and semilattice types (+ some extra conditions) ongoing, many people: languages that admit the semilattice and affine types

22. Dichotomy results Schaefer 1978:  over a 2-element set Hell, Nesetril 1990:  = {E}, where E is binary symmetric • 2006:  over a 3-element set Markovic, McKenzie >2011:  over a 4-element case 1 case out of left (as of last Wednesday) B. 2003, Barto 2011:  conservative, that is, it contains all unary relations

23. Polymorphisms of conservative languages If is a polymorphism of a conservative language , then for any We look at how polymorphisms behave on 2-element subsets If for some 2-elemen subset B there is no polymorphism that is good on B then CSP() is NP-complete Theorem (B. 2003) CSP() for a conservative  on A is poly time iff for any 2-element B  A there is f  Pol() which is affine, majority, or semilattice; otherwise CSP() is NP-complete.

24. semilattice operation majority operation affine operation Edge coloured graphs G(): Since semilattice operation induces an order, red edges are directed

25. AS-components Let  be a conservative language over set A B  A is called an as-component (affine-semilattice) if it is minimal with respect to the property: there is no affine or semilattice (directed) edge in G() sticking out of B The remaining edges are majority

26. Chinese Remainder Theorem for AS-Component Let R  for a conservative  on A and as-components such that for any i,j  {1,...,k} there is a tuple such that Then there is such that for all i,j  {1,...,k}. CRT for AS-Components

27. Coherent Sets Let R be a k-ary relation on A, let be as-components Positions i and j are -related if for any iff I  {1,...,k} is a coherent set w.r.t. as-components if any i,j  I are -related

28. Rectangularity Lemma Let R  and as-components such that Let also be the partition of {1,...,k} into coherent sets w.r.t. and Then Rectangularity

29. u - v - w - x - y - The Algorithm • On input (V,D,C) • run 2-consistency algorithm • find as-components such that for any v,w  V as-components • are consistent • find the coherent sets • for each coherent set W solve the problem restricted to W and • if every such problem has a solution, any combination of such solutions gives a solution to the problem • otherwise remove elements the failed as-components and start over

30. General Case I Ingredients: the graph Chinese Remainder Theorem rectangularity solving smaller problems failed components removal

31. General Case II The graph Cannot get a complete graph, but connected is possible Semilattice and majority edges are defined in almost the same way: a,b if there is a polymorphism which is semilattice or majority on {a,b} Affine edges: Instead of pairs of elements use subset B A and a partition of B such that there is a polymorphism that acts as an affine operation on the set B A

32. General Case III Chinese Remainder Theorem: holds Rectangularity: does not hold, need a weaker condition Solving smaller problems Theorem There is a poly time algorithm such that on (V,A,C) - if for each v  V and any element a from an as-component there is a solution  with (v) = a, then the algorithm finds a solution; - otherwise it identifies which elements from as-components are not a part of a solution.

33. u - v - w - x - y - General Case IV Failed components removal Have to check every element if it is a part of a solution, not only maximal ones

34. Conclusion Current Score: 3 : 2 Thank you!