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
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
7.
8.
9.
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
15. Good Polymorphisms: Majority Propagation again: 2-consistency Any 2-consistent instance has a solution u - v - w - x - y -
16.
17.
18.
19.
20.
21.
22.
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
25.
26. 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 }.
27.
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
