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Detecting and Exploiting Subproblem Tractability Christian Bessiere, Cl ement Carbonnel, Emmanuel Hebrard, George Katsirelos and Toby Walsh August 6th 2013 Emmanuel Hebrard () Subproblem Tractability August 6th 2013 1 / 16 Tractability

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Detecting and Exploiting Subproblem Tractability

Christian Bessiere, Cl´ ement Carbonnel, Emmanuel Hebrard, George Katsirelos and Toby Walsh August 6th 2013

Emmanuel Hebrard () Subproblem Tractability August 6th 2013 1 / 16

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Tractability

Lots of research on tractable constraint problems

Restricted language (e.g. 2SAT) Restricted constraint structure (e.g. tree)

But solvers often perform poorly on tractable problems

Not enough to know it is tractable [Petke & Jeavons 2009] Detect membership to a tractable and apply the proper algorithm

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Tractability

Lots of research on tractable constraint problems

Restricted language (e.g. 2SAT) Restricted constraint structure (e.g. tree)

But solvers often perform poorly on tractable problems

Not enough to know it is tractable [Petke & Jeavons 2009] Detect membership to a tractable and apply the proper algorithm

Problems might be nearly-tractable

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Near-Tractability

a b c e f g d Solid edges: “easy” constraints / Dashed edges: “hard” constraints

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Near-Tractability

a b c e f g d = v Solid edges: “easy” constraints / Dashed edges: “hard” constraints Branch on d

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Near-Tractability

a b c e f g Solid edges: “easy” constraints / Dashed edges: “hard” constraints Branch on d

Remove (a, d) and (d, g)

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Motivation

x y y 1 1 1 P1 P2 P3 P4 Assigned variables Tractable CSPs

Identify a (hopefully) small number of variables Branch on these to give tractable subproblems

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Motivation

x y y 1 1 1 P1 P2 P3 P4 Assigned variables Tractable CSPs

Identify a (hopefully) small number of variables Branch on these to give tractable subproblems find a backdoor [Williams et al. 2003]

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Contributions

Detecting tractable classes Exploiting tractable classes

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Contributions

Detecting tractable classes

Detecting set of relations closed by a majority polymorphism Detecting set of relations closed by a Mal’tsev polymorphism

Exploiting tractable classes

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Contributions

Detecting tractable classes

Detecting set of relations closed by a majority polymorphism Detecting set of relations closed by a Mal’tsev polymorphism

Exploiting tractable classes

If given a tractable sublanguage: easy Otherwise: hard (but there are positive results!)

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Polymorphisms and Tractability

Constraint problems are tractable if their relations are closed under majority polymorphisms [Jeavons et al 1997]

Generalization of 2-SAT and 0/1/all constraints

Constraint problems are tractable if their relations are closed under Mal’tsev polymorphisms [Bulatov & Dalmau 2006]

Generalization of linear equations over a field

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm)

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

Emmanuel Hebrard () Subproblem Tractability August 6th 2013 7 / 16

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Polymorphism

Operation f maps m values v1, . . . , vm to another value f (v1, . . . , vm) Similarly it maps m tuples τ1, . . . , τm to another tuple f (τ1, . . . , τm) f is a polymorphism of R iff applying f to tuples of R does not produce new tuples Example f (x, y) = (x + y mod 2) 1 1 1 1 1 1 R :

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Detecting Majority

f is a majority operation iff f (v, v, w) = f (v, w, v) = f (w, v, v) = v

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Detecting Majority

f is a majority operation iff f (v, v, w) = f (v, w, v) = f (w, v, v) = v Theorem 1 Majority polymorphisms can be detected in polynomial time

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Detecting Majority

f is a majority operation iff f (v, v, w) = f (v, w, v) = f (w, v, v) = v Polymorphisms of P are solutions of its indicator problem [Jeavons et

al. 1997]

P and its indicator problem share the same set of relations

A CSP closed under a majority polymorphism is solved backtrack-free by maintaining singleton arc consistency [Chen et al. 2013] Theorem 1 Majority polymorphisms can be detected in polynomial time

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Detecting Majority

f is a majority operation iff f (v, v, w) = f (v, w, v) = f (w, v, v) = v Polymorphisms of P are solutions of its indicator problem [Jeavons et

al. 1997]

P and its indicator problem share the same set of relations

A CSP closed under a majority polymorphism is solved backtrack-free by maintaining singleton arc consistency [Chen et al. 2013] Theorem 1 Majority polymorphisms can be detected in polynomial time Proof Run maintain SAC on the indicator problem

If success, we obtain a solution, hence a polymorphism of P If at any point there is a fail, we can deduce that the indicator problem has no majority polymorphism

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Detecting (conservative) Mal’tsev

f is a Mal’tsev operation iff f (v, w, w) = f (w, w, v) = v f is conservative iff f (u, v, w) ∈ {u, v, w} Theorem 2 Conservative Mal’tsev polymorphisms can be detected in polynomial time

n binary relations

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Exploiting Tractability

Input: A CSP P = (X, D, C), a set B such that C \ B has a polymorphism Question: is P satisfiable?

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Exploiting Tractability

Input: A CSP P = (X, D, C), a set B such that C \ B has a polymorphism Question: is P satisfiable?

Backdoor of size k: search tree of size dk

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Exploiting Tractability

Input: A CSP P = (X, D, C), a set B such that C \ B has a polymorphism Question: is P satisfiable?

Backdoor of size k: search tree of size dk

Fixed Parameter Tractability Given a problem A and a parameter k Fixed Parameter Tractable (FPT) iff there exists an algorithm which complexity is in O(f (k)P(n))

Any computable function f of k (for ex. 2k) A polynomial P(n) in the size of the problem n

Emmanuel Hebrard () Subproblem Tractability August 6th 2013 10 / 16

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Exploiting Tractability

Input: A CSP P = (X, D, C), a set B such that C \ B has a polymorphism Question: is P satisfiable?

Backdoor of size k: search tree of size dk

Fixed Parameter Tractability Given a problem A and a parameter k Fixed Parameter Tractable (FPT) iff there exists an algorithm which complexity is in O(f (k)P(n))

Any computable function f of k (for ex. 2k) A polynomial P(n) in the size of the problem n

W [m]-hardness by reduction from a W [m]-hard pair A′, k′

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