On Guaranteeing Polynomially Bounded Search Tree Size D.Cohen 1 - PowerPoint PPT Presentation

On Guaranteeing Polynomially Bounded Search Tree Size D.Cohen 1 M.Cooper 2 M.Green 1 D.Marx 3 Dept. of Computer Science, Royal Holloway, University of London, UK IRIT, University of Toulouse III, 31062 Toulouse, France Institut für Informatik, Humboldt-Universität zu Berlin, Germany The 17th International Conference on Principles and Practice of Constraint Programming, 2011

Outline Background 1 2 Motivation 3 (Incrementally) Function 4 k -Turan 5 Where now?

Two reasons for tractability small representation of the (partial) solution sets.

Two reasons for tractability small representation of the (partial) solution sets. A local (bounded width) decision procedure.

Two reasons for tractability small representation of the (partial) solution sets. A local (bounded width) decision procedure. Example Linear Equations. Bounded treewidth, Max-closed languages.

Small representations How? How can we find new tractable classes which are tractable because their solution sets have a small representation?

Small representations How? How can we find new tractable classes which are tractable because their solution sets have a small representation? Its Obvious Polynomially many solutions!

Small representations How? How can we find new tractable classes which are tractable because their solution sets have a small representation? Its Obvious Polynomially many solutions! . . . but Does not work. Such constraint networks are not recognisable

A General Framework Theorem Let P = � V , D , C � be any constraint network with variable ordering x 1 < x 2 < . . . < x n such that the induced network P [ { x 1 , . . . , x i } ] can be solved in time p ( | P [ { x 1 , . . . , x i } ] | ) . All solutions to P can be enumerated in time p ( | P | ) . | P | 2 .

A General Framework Theorem Let P = � V , D , C � be any constraint network with variable ordering x 1 < x 2 < . . . < x n such that the induced network P [ { x 1 , . . . , x i } ] can be solved in time p ( | P [ { x 1 , . . . , x i } ] | ) . All solutions to P can be enumerated in time p ( | P | ) . | P | 2 . . . . and a well-known example Grohe, Marx (2006): The number of solutions to a constraint network of size A is at most A w , where w is the width of a fractional edge cover. This works because an induced network cannot increase the required fractional edge cover.

Functional Definition A constraint � σ, ρ � is functional on variable i ∈ σ if ρ contains no two tuples differing only at variable i . A constraint network P is functional if there exists a variable ordering such that, for all i ∈ { 1 , . . . , n } , there is some constraint of P [ { x 1 , . . . , x i } ] that is functional on x i .

Functional Definition A constraint � σ, ρ � is functional on variable i ∈ σ if ρ contains no two tuples differing only at variable i . A constraint network P is functional if there exists a variable ordering such that, for all i ∈ { 1 , . . . , n } , there is some constraint of P [ { x 1 , . . . , x i } ] that is functional on x i . Example Variables: { x 1 , . . . , x 4 } . Domain: Integers modulo 7 ( { 0 , . . . , 6 } ). Constraints: x 1 = 4 , 2 x 1 + x 2 = 5 , 3 x 1 + 4 x 2 + x 3 = 2 and x 1 + x 2 + x 3 + x 4 = 0. Ordering: x 1 < x 2 < x 3 < x 4 . Solution (unique): x 1 = 4 , x 2 = 4 , x 3 = 2 , x 4 = 4.

. . . and a new one Definition A constraint network P is incrementally functional if there is an ordering of its variables such that for all i ∈ { 1 , . . . , n − 1 } , each solution to P [ { x 1 , . . . , x i } ] extends to at most one solution to P [ { x 1 , . . . , x i + 1 } ] .

. . . and a new one Definition A constraint network P is incrementally functional if there is an ordering of its variables such that for all i ∈ { 1 , . . . , n − 1 } , each solution to P [ { x 1 , . . . , x i } ] extends to at most one solution to P [ { x 1 , . . . , x i + 1 } ] . Theorem Given a constraint network P = � V , D , C � , it is possible to find in polynomial time the maximum-cardinality set M ⊆ V such that P [ M ] is incrementally functional.

. . . and a new one Definition A constraint network P is incrementally functional if there is an ordering of its variables such that for all i ∈ { 1 , . . . , n − 1 } , each solution to P [ { x 1 , . . . , x i } ] extends to at most one solution to P [ { x 1 , . . . , x i + 1 } ] . logarithmically close The class of networks which are “nearly” incrementally functional is tractable.

. . . and a surprising negative one Definition In a constraint network P = � V , D , C � , a root set is a subset Q of the variables for which there exists a variable ordering x 1 < x 2 < . . . < x n such that, for all i ∈ V − Q , there is some constraint of P [ { x 1 , . . . , x i } ] that is functional on x i .

. . . and a surprising negative one Definition In a constraint network P = � V , D , C � , a root set is a subset Q of the variables for which there exists a variable ordering x 1 < x 2 < . . . < x n such that, for all i ∈ V − Q , there is some constraint of P [ { x 1 , . . . , x i } ] that is functional on x i . Theorem The problem of finding a minimum-cardinality root set in a ternary constraint network is NP-hard. Proven by a reduction from MAX-CLIQUE

. . . and a surprising negative one Definition In a constraint network P = � V , D , C � , a root set is a subset Q of the variables for which there exists a variable ordering x 1 < x 2 < . . . < x n such that, for all i ∈ V − Q , there is some constraint of P [ { x 1 , . . . , x i } ] that is functional on x i . logarithmically close The class of networks which are “nearly” incrementally functional is tractable. The (smaller) class of networks with a “small” root set is (probably) not tractable.

Can we do it any other way? Is this the end - my friend Is there another way - other than by the use of functional constraints or fractional edge covers?

Can we do it any other way? Is this the end - my friend Is there another way - other than by the use of functional constraints or fractional edge covers? . . . another well-known result A k -SAT instance in which every set of k variables occurs in a clause has at most n k satisfying assignments.

k -Turan Definition A set σ represents another set τ if σ is contained in τ . An ( n , k ) -Turan system is a collection of subsets of the n -element set χ representing every k -element subset of χ .

k -Turan Definition A set σ represents another set τ if σ is contained in τ . An ( n , k ) -Turan system is a collection of subsets of the n -element set χ representing every k -element subset of χ . Definition An n -variable constraint network over domain D is said to be k -Turan if the scopes of the constraints � σ, ρ � for which: ∀ a , b ∈ D , [ a , b ] | σ | �⊆ ρ are an ( n , k ) -Turan system.

k -Turan tractability Theorem For any domain D and fixed k, the class of k-Turan constraint networks is tractable.

k -Turan examples Example (Folklore, Mists of Time) The class of k -SAT instances where every k -tuple is restricted by a k -clause.

k -Turan examples Example (Folklore, Mists of Time) The class of k -SAT instances where every k -tuple is restricted by a k -clause. Example (David, 1995) Constraint networks that have a functional constraint for every scope of arity k .

k -Turan examples Example (Folklore, Mists of Time) The class of k -SAT instances where every k -tuple is restricted by a k -clause. Example (David, 1995) Constraint networks that have a functional constraint for every scope of arity k . Example (NEW!) Variables: { 1 , . . . , n } . Scopes: pairs with the same parity. Constraints: only disallow d − 1 tuples - � 2 , 2 � , . . . , � d , d � .

Where now? (Strong) Polynomially many Solutions

Where now? (Strong) Polynomially many Solutions Structural: Fractional edge covers of weight at most k . Very large scopes, no limit on the relations.

Where now? (Strong) Polynomially many Solutions Structural: Fractional edge covers of weight at most k . Very large scopes, no limit on the relations. Hybrid: Nearly incrementally functional. Functional constraints are realistic. Nice Theory.

Where now? (Strong) Polynomially many Solutions Structural: Fractional edge covers of weight at most k . Very large scopes, no limit on the relations. Hybrid: Nearly incrementally functional. Functional constraints are realistic. Nice Theory. Hybrid: k -Turan for a fixed domain. Constraints are realistic. Scopes need not be large. Needs many scopes.

Where now? (Strong) Polynomially many Solutions Structural: Fractional edge covers of weight at most k . Very large scopes, no limit on the relations. Hybrid: Nearly incrementally functional. Functional constraints are realistic. Nice Theory. Hybrid: k -Turan for a fixed domain. Constraints are realistic. Scopes need not be large. Needs many scopes. . . .