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CSP: Algorithms and Dichotomy Conjecture Andrei A. Bulatov Simon Fraser University
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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
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Constraint Satisfaction Problem II u - v - w - x - y - Q ( u,v,w ) R ( w,x ) R ( x,y ) S ( y,u )
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Examples: Linear Equations, SAT Linear Equations : 3-SAT = CSP( ) :
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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
does not have any polymorphisms except for very trivial ones, e.g. f ( x,y,z ) =y
Polymorphisms: 3-COL
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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
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
Good Polymorphisms: Majority 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
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Good Polymorphisms: Majority Propagation again: 2-consistency Any 2-consistent instance has a solution u - v - w - x - y -
Complexity: Boolean CSP 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
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
Complexity: Graphs 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
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
Types 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
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
Dichotomy results
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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.
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Edge coloured graphs G ( ) : Since semilattice operation induces an order, red edges are directed semilattice operation majority operation affine operation
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
AS-components The remaining edges are majority
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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 }.
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
Coherent Sets
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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
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
Rectangularity : does not hold, need a weaker condition
Solving smaller problems
General Case III 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.
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