FPT-reduction ( P , k ) is fpt-reducible to ( P ′ , k ′ ) if there is → { inputs to P ′ } R : { inputs to P } − ⇒ R ( x ) ∈ P ′ x ∈ P ⇐ given x , problem R ( x ) can be solved in FPT-time k ′ ( R ( x )) ≤ g ( k ( x )) for some computable g
W-hierarchy ( P , k ) ∈ W [1] if it is FPT-reducible to finding a weight k satisfying assignment to a boolean circuit of depth 1.
W-hierarchy FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . .
W-hierarchy FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . . p -Clique is W[1]-complete under FPT-reductions.
W-hierarchy FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . . p -Clique is FPT-reducible to p -HOM( C ,–) Does G have a Is there a homomorphism k -clique? K k − → G ?
W-hierarchy FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . . p -HOM( C ,–) is W[1]-hard if C contains all complete graphs
Main theorem Assume FPT � = W[1]. For every recursively enumerable class C of relational structures of bounded arity the following statements are equivalent 1. HOM( C ,–) is in polynomial time. 2. p -HOM( C ,–) is fixed-parameter tractable. 3. C has bounded treewidth modulo homomorphic equivalence.
Main theorem Assume FPT � = W[1]. Dalmau et al, 2002. For every recursively enumerable class C of relational structures of bounded arity the following statements are equivalent ( w + 1) pebble-game 1. HOM( C ,–) is in polynomial time. 2. p -HOM( C ,–) is fixed-parameter tractable. 3. C has bounded treewidth modulo homomorphic equivalence.
Main theorem Assume FPT � = W[1]. For every recursively enumerable class C of relational structures of bounded arity p -HOM(C,–) is not W[1]-hard. the following statements are equivalent p -Clique to p -HOM(C,–) reduction. 1. HOM( C ,–) is in polynomial time. 2. p -HOM( C ,–) is fixed-parameter tractable. 3. C has bounded treewidth modulo homomorphic equivalence.
Main theorem Assume FPT � = W[1]. For every recursively enumerable class C of relational structures of bounded arity p -HOM(C,–) is not W[1]-hard. the following statements are equivalent 1. HOM( C ,–) is in polynomial time. 2. p -HOM( C ,–) is fixed-parameter tractable. 3. C has bounded treewidth modulo homomorphic equivalence.
Constraint satisfaction problems A CSP is a triplet ( V , D , C ) V : variables D : domain of the variables C : constraints
A 3-SAT formula as a CSP ( a ∨ b ∨ c ) ∧ ( b ∨ c ∨ d ) ∧ ( a ∨ d ) V = { a , b , c , d } D = { T , F } C : a ∨ b ∨ c = { ( T , T , T ) , ( T , T , F ) , ( T , F , T ) , . . . } b ∨ c ∨ d = { ( T , T , T ) , ( T , T , F ) , ( T , F , T ) , . . . } a ∨ d = { ( T , T ) , ( T , F ) , ( F , T ) }
Constraint satisfaction problems Constraint Satisfaction Instance: a CSP ( V , D , C ) Problem: is there an assignment V − → D satisfying all constraints in C ?
Constraint satisfaction problems I = ( V , D , C ) Hypergraph Universe: V Relations: scopes of relations in C
Constraint satisfaction problems I = ( V , D , C ) Hypergraph Constraints Universe: V Universe: D Relations: scopes of Relations: actual relations in C relations in C
Constraint satisfaction problems I = ( V , D , C ) Hypergraph Constraints Universe: V Universe: D Relations: scopes of Relations: actual relations in C relations in C ?
CSP satisfiability as homomorphism problem ( a ∨ b ∨ c ) ∧ ( b ∨ c ∨ d ) ∧ ( a ∨ d )
CSP satisfiability as homomorphism problem ( a ∨ b ∨ c ) ∧ ( b ∨ c ∨ d ) ∧ ( a ∨ d ) A A = { a , b , c , d } R A = { ( a , b , c ) } ⊆ A 3 S A = { ( b , c , d ) } ⊆ A 3 T A = { ( a , d ) } ⊆ A 2
CSP satisfiability as homomorphism problem ( a ∨ b ∨ c ) ∧ ( b ∨ c ∨ d ) ∧ ( a ∨ d ) A B A = { a , b , c , d } B = { T , F } R A = { ( a , b , c ) } ⊆ A 3 R B = { ( T , T , T ) , . . . } S A = { ( b , c , d ) } ⊆ A 3 S B = { ( T , T , T ) , . . . } T A = { ( a , d ) } ⊆ A 2 T B = { ( T , T ) , . . . }
CSP result Assume FPT � = W[1]. For every recursively enumerable class C of relational structures of bounded arity the following statements are equivalent 1. HOM( C ,–) is in polynomial time. 2. p -HOM( C ,–) is fixed-parameter tractable. 3. C has bounded treewidth modulo homomorphic equivalence.
CSP result Assume FPT � = W[1]. For every recursively enumerable class C of graphs the following statements are equivalent 1. HOM( C ,–) is in polynomial time. 2. p -HOM( C ,–) is fixed-parameter tractable. 3. C has bounded treewidth modulo homomorphic equivalence.
CSP result Assume FPT � = W[1]. For every recursively enumerable class C of graphs the following statements are equivalent 1. CSP( C ) is in polynomial time. 3. C has bounded treewidth modulo homomorphic equivalence.
A subexponentional lower bound CSPs of bounded treewidth k can be solved in time n O ( k )
A subexponentional lower bound If there exists a recursively enumerable class G of graphs with unbounded treewidth, and a computable function f such that CSP( G ) can be solved in time f ( G ) · | I | o ( tw ( G ) / log tw ( G )) then ETH fails. Marx 2007, Can we beat treewidth?
Probabilistic Inference Positive Inference Instance: a Bayesian network, evidence e , query variable x Problem: does Pr ( x | e ) > 0 hold?
The necessity of bounded treewidth If there exists a computable function f such that Positive Inference can be decided in f ( G ) · | B | o ( tw ( G ) / log tw ( G )) G is the moralized graph of the Bayesian network B , then ETH fails.
The necessity of bounded treewidth Proof. Reduce Constraint Satisfaction to Positive Inference preserving treewidth.
TW-reducibility A is tw-reducible to B if there exists a polynomial-time computable function g and a linear function l such that x ∈ A ⇐ ⇒ g ( x ) ∈ B tw ( g ( x )) = l ( tw ( x ))
TW reduction of CSP to Bayesian network X 1 X 2 X 4 X 3
TW reduction of CSP to Bayesian network R 1 R 4 X 1 X 2 X 4 X 3 R 2 R 3
TW reduction of CSP to Bayesian network R 1 R 4 X 1 X 2 X 4 A X 3 R 2 R 3
TW reduction of CSP to Bayesian network R 1 R 4 X 1 X 2 X 4 X 3 R 2 R 3
TW-reduction of CSP to Bayesian network R 1 R 4 X 1 X 2 X 4 X 3 R 2 R 3
TW-reduction of CSP to Bayesian network R 1 R 4 X 1 X 1 X 2 X 4 X 6 X 2 X 4 X 3 X 5 X 3 R 2 R 3
TW-reduction of CSP to Bayesian network R 1 R 4 X 1 X 1 X 2 X 4 X 6 X 2 X 4 X 3 X 5 X 3 R 2 R 3
TW-reduction of CSP to Bayesian network R 1 R 4 X 1 X 1 X 2 X 4 X 6 X 2 X 4 X 3 X 5 X 3 R 2 R 3
TW-reduction of CSP to Bayesian network R 1 R 4 X 1 X 1 X 2 X 4 X 6 X 2 X 4 X 3 X 5 X 3 R 2 R 3
TW-reduction of CSP to Bayesian network R 1 R 4 X 1 X 1 X 2 X 4 X 6 X 2 X 4 X 3 X 5 X 3 R 2 R 3
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