The document discusses the constraint satisfaction problem (CSP) and the dichotomy conjecture in computational complexity theory. It defines CSP and provides examples. It discusses the role of polymorphisms - operations that preserve constraints. The presence or absence of certain polymorphisms like semilattice, majority, and affine operations determines the complexity of CSP for a given constraint language. The document outlines a proposed dichotomy - CSP is either solvable in polynomial time or NP-complete, depending on the polymorphisms. It surveys partial results proving this conjecture and algorithms for certain constraint languages.

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

2. Constraint Satisfaction Problem I CSP(  ) 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 

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

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

6. Invariants and Polymorphisms Pol(  ) denotes the set of all polymorphisms of relations from  Definition A relation 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

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

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

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

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

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

27. CRT for AS-Components 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 }.

28. Rectangularity 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