Nov

10 20091

Properties of Tree Convex Constraints

Authors: Yuanlin Zhang & Eugene C. Freuder

Presentation by Robert J. WoodwardCSCE990 ACP, Fall 2009

Nov

10 20092

Overview• Introduction• Definitions

– Basic, support, image, consistency, trees, tree intersection, tree convex constraint, tree convex CSPs,

• Properties– Intersection & composition on tree convex constraints– Consecutiveness: definition & composition

• Tractable Networks– Locally Chain Convex– Local Chain Convex & Strictly Union Closed (LCC&SUC)

• (One) Application• Related Work• Conclusion

Nov

10 20093

Story• Introduces tree convexity

PC tree convexity global consistency• PC uses the operators ◦ and

– ◦ ‘damages’ the tree convexity property• Introduces ‘consecutiveness’ property, which is closed under

◦… but not under • Introduces ‘locally chain convex’ (LCC) property, which is

closed when domains are filtered, but is not closed under ◦• Introduces ‘locally chain convex & strictly union closed’

(LCC&SUC) which is closed under ◦ and PC LCC&SUC global consistency

Nov

10 20094

Introduction

• Binary constraint network

• Tree Convex– Construct a tree for a

variables domain– All allowed supports

must form subtree

a b c da 1 1 1 0b 1 0 1 1c 1 1 0 1

a

b c d

{a,b,c,d}{a,b,c}

x y

cxy

cxy =

Tree for y

• Linear-time algorithm can detect tree convexity

[Conitzer+ AAAI04]

Nov

10 20095

Introduction

• Binary constraint network

• Tree Convex– Construct a tree for a

variables domain– All allowed supports

must form subtree

b c da 1 1 0b 0 1 1c 1 0 1

b

c d

{b,c,d}{a,b,c}

x y

cxy

cxy =

Tree for y

• Linear-time algorithm can detect tree convexity

[Conitzer+ AAAI04]

Nov

10 20096

Definitions: Basic

• Constraint networks (Binary)– Variables: V = {x1,x2,…,xn}– Domains: Di for each xi ϵ V. Finite– Constraints: Between ordered variables

• Constraint between (x,y) is cxy

• cyx is a different constraint

– Operations (on constraints)• Intersection (∩)• Composition (◦)• Inverse

Nov

10 20097

Definitions: Support, Image

• For a constraint cxy

• Value u ϵ Dx

• Value v ϵ Dy

– support: u and v satisfy cxy

– image of u under cxy

• denoted Iy(u)• Set of supports in Dy

– image of a subset of Dx

• Union of images of its values

a b c da 1 1 1 0b 1 0 1 1c 1 1 0 1

{a,b,c,d}{a,b,c}

x y

cxy

cxy =

•What is Iy(b)?• {a,c,d}

Nov

10 20098

Definitions: Basic Consistency

• k-consistency– Any distinct k-1 variables can be consistently

extended to another• Strongly k-consistent– j-consistent for all j≤k– Globally consistent: strongly n-consistent– Strongly 2-consistency: arc consistent– Strongly 3-consistency: path consistent

Nov

10 20099

Definitions: Trees• Tree

– Connected graph without any cycles

– Path between any two nodes is unique

– Distance of a node to the root is the number of edges in the path

– Subtree is a connected subgraph of the tree.• Root is the node closed to root of

tree

• Chain– At most one child per node– Last value of chain furthest away

from root

• Forest– Graph without any cycles– Can also be looked at as a set

of trees– Assume root for a tree in a

forest• Forest on a set S

– vertex set is exactly S• Set I is subtree of a forest

– If there exists a subtree of some tree whose vertex set is exactly I

– Note: Ø subtree of any forest

Nov

10 200910

Definitions: Tree Intersection

• Intersection of two trees on a common tree is a tree whose vertices and edges are in both trees

• Proposition 1: Intersection of two trees is a subtree of the tree– If the intersection is not

empty, root of intersection is root of one of the trees

a

b d

e fx

a

b c

∩

x

y za

b c

Common Tree:

d

Nov

10 200911

Definitions: Tree Intersection

• Intersection of two trees on a common tree is a tree whose vertices and edges are in both trees

• Proposition 1: Intersection of two trees is a subtree of the tree– If the intersection is not

empty, root of intersection is root of one of the trees

a

b d

e fx

a

b c

∩

x

y za

b c

Common Tree:

d

Nov

10 200912

Definitions: Tree Convex

• Definition 1– Sets E1,…,Ek are tree

convex with respect to a forest T on Uiϵ1..kEi

– If every Ei is a subtree of T

Sets that are tree convex• {a,b,c}• {a,b,d}• {a,d}• {a,c}• {d}• …

a

b c d

Nov

10 200913

Definitions: Tree Convexity on cxy

• Definition 2– A constraint cxy is tree

convex with respect to a forest T on Dy if the images of all values of Dx are tree convex with respect to T

a b c da 1 1 1 0b 1 0 1 1c 1 1 0 1

a

b c d

{a,b,c,d}{a,b,c}

x y

cxy

cxy =

Tree for y

Nov

10 200914

• A CSP is tree convex– If there exists one forest for every variable &– Every constraint in CSP is tree convex with respect to this forest

• Tree convex CSP is globally consistent if it is path consistent– Proof in [Zhang & Yap, IJCAI 2003]– Let R be a network of constraints with arity at most r and R be

strongly 2(r-1)+1 consistent. If R is tree convex then it is globally consistent• For binary constraints• strongly 2(2-1)+1 consistent = path consistent

Definitions: Tree Convex CSP

Nov

10 200915

Definition: Proof• Tree convex constraint network

is globally consistent if it is path consistent– Network is path consistent– Prove by induction k consistent

• k ϵ {4,…,n}– Consider instantiation of any k-1

variables and any new variable x– Number of relevant constraints be l• For each relevant constraint there is

one extension to x– We have l extension sets.– If intersection of all l sets is not

empty, x satisfies all relevant constraints

– Consider two of the l extension sets, E1 and E2

– Consistency lemma• if network is path consistent• Ix(E1)∩Ix(E2) Ø

– Since all constraints in l are tree convex• Extension sets are tree convex

– Tree convex sets intersection lemma• ∩Ix(EiϵL) Ø iff Ix(Ej)∩Ix(Ek) Ø

– From consistency lemma, we have k-consistent• Network is k+1 consistent iff any

instantiation of k distinct variables and a new variable, ∩Ic(EiϵL) Ø

• Therefore, by induction– globally consistent

[Zhang &Yap 03]

Nov

10 200916

Overview• Introduction• Definitions

– Basic, support, image, consistency, trees, tree intersection, tree convex constraint, tree convex CSPs

• Properties– Intersection & composition on tree convex constraints– Consecutiveness: definition & composition

• Tractable Networks– Locally Chain Convex– Local Chain Convex & Strictly Union Closed (LCC&SUC)

• (One) Application• Related Work• Conclusion

Nov

10 2009

a

b

c d

Properties: Intersection & Composition

• Assume c1xy c2

xy are tree convex to a forest T on domain Dy

• Prove there intersection is tree convex– Let cxy = c1

xy c2xy

– For any v in Dx

• images under c1xy and c2

xy are subtrees of T

• intersection of two images is a subtree of T by Proposition 1

– Every image of every v in Dx is a subtree of T

– Therefore, cxy is tree convex

• Composition does not preserve tree convexity

a

b

c d

x y

c2xy

x y

c1xy

c1xy

c2xy

17

Nov

10 200918

Properties: Consecutiveness• Consecutive

– Tree convex constraint cxy with respect to a forest Ty on Dy is consecutive with respect to a forest Tx on Dx

– iff every two neighboring values a,b on Tx, Iy(a) U Iy(b) is subtree of Ty

• Constraint Network tree convex and consecutive– exists a forest on each domain– every constraint cxy is tree convex

and consecutive with respect to the forests on Dy and Dx

a b c da 1 1 1 0b 1 0 1 0c 1 1 0 1

a

b c d

{a,b,c,d}{a,b,c}

x y

cxy

cxy =

Tree for y

a

b cTree for x

{a,b,c}{a,b,c,d}

Nov

10 200919

Properties: Consecutiveness composition

• Class of consecutive tree convex constraints is closed under composition– Let cxy and cyz be two consecutive tree

convex constraints with trees Tx, Ty, and Tz

– cxz is their composition– cxz is tree convex

• v in Dx

• image under cxz = UbϵIy(v)Iz(b)

• Union of images of neighboring values in Iy(v) is subtree of Tz, union of all values in Iy(v) is a subtree too

– cxz is consecutive• u,v in Dx and neighbors under Tx

• Since cxy is consecutive, Iy(u) U Iy(v) is subtree to Ty

• Iy(u) U Iy(v) is also a subtree of Tz because of consecutiveness of Cyz

• So, Iz(u) U Iz(v) is a subtree of Tz

a

b

c d

x

a

b

c d

y

a

b

c d

z

cxy cyz

Nov

10 200920

Properties: Consecutiveness composition

• Class of consecutive tree convex constraints is closed under composition– Let cxy and cyz be two consecutive tree

convex constraints with trees Tx, Ty, and Tz

– cxz is their composition– cxz is tree convex

• v in Dx

• image under cxz = UbϵIy(v)Iz(b)

• Union of images of neighboring values in Iy(v) is subtree of Tz, union of all values in Iy(v) is a subtree too

– cxz is consecutive• u,v in Dx and neighbors under Tx

• Since cxy is consecutive, Iy(u) U Iy(v) is subtree to Ty

• Iy(u) U Iy(v) is also a subtree of Tz because of consecutiveness of Cyz

• So, Iz(u) U Iz(v) is a subtree of Tz

a

b

c d

x

a

b

c d

y

a

b

c d

z

cxy cyz

Nov

10 200921

Overview• Introduction• Definitions

– Basic, support, image, consistency, trees, tree intersection, tree convex constraint, tree convex CSPs

• Properties– Intersection & composition on tree convex constraints– Consecutiveness: definition & composition

• Tractable Networks– Locally Chain Convex– Local Chain Convex & Strictly Union Closed (LCC&SUC)

• (One) Application• Related Work• Conclusion

Nov

10 200922

Tractable Networks: Locally Chain Convex

• Intersection of two subtrees could be empty– Image of value could be empty– Deleting value makes constraint not

tree convex• Locally chain convex

– Constraint cxy with respect to a forest on Dy

– iff image of every value in Dx is subchain of forest

• Constraint network locally chain convex– iff exists forest on each domain such

that every constraint is locally chain convex

a b c da 1 1 1 0b 1 0 1 1c 1 1 0 1

a

b c d

{a,b,c,d}{a,b,c}

x y

cxy

cxy =

Tree for y

Nov

10 200923

Tractable Networks: Locally Chain Convex

• Locally chain convex constraint network is locally chain convex after removal of any value from domain– Consider a variable y

• Assume forest on Dy is Ty

– Value v is removed from Dy

– Need to show every cxy in C is locally chain convex• Could have made images on Dx not connected

• Construct new forest Ty’’ on Dy

– broken subchains will be reconnected

Nov

10 200924

Tractable Networks: Locally Chain Convex

• Let v1,…,vl be children of v• Let pv be parent of v• Construct a new forest Ty’

from Ty

• Remove v and all edges incident on v

• Construct Ty’’ from Ty’– Add edge between pv and all vi

• If v is root of Ty, let Ty’’ be Ty’

pv

v

v1v2

pv

v

v1v2 v1v2

pv

Ty’ Ty’’Ty

Nov

10 200925

Tractable Networks: Locally Chain Convex

• Let v1,…,vl be children of v• Let pv be parent of v• Construct a new forest Ty’

from Ty

• Remove v and all edges incident on v

• Construct Ty’’ from Ty’– Add edge between pv and all vi

• If v is root of Ty, let Ty’’ be Ty’

Ty’ Ty’’

v

v1v2

Ty

v

v1v2 v1v2

Nov

10 200926

Tractable Networks: Locally Chain Convex

• Composition may destroy local chain convexity

• To get tractable class we need to combine– Local chain convexity

• For deleting a value– Consecutiveness

• For composition

a

bc d

a

bc d

a

bc d

x y z

cyzcxy

a

bc d

a

bc d

x z

cxz

(Composition)

Nov

10 200927

Tractable Networks: LCC&SUC• Locally chain convex and

strictly union closed (LCC&SUC)– With respect to forest Tx on Dx

and Ty on Dy

– image of any subchain in Tx is subchain in Ty

• Constraint network is locally chain convex and strictly union closed– every constraint cxy is locally

chain convex and strictly union closed with respect to the forests Dx and Dy

a

bc d

a

bc d

a

bc d

x y z

cyzcxy

•Consider subchain {a,b} in y. What are images in z?• {b,c}, it is a subchain

•Consider subchain {b,d} in y. What are images in z?• {b,c,d}, it is not a subchain

Nov

10 200928

Tractable Networks: LCC&SUC

• A locally chain convex and strictly union closed constraint network can be transformed to an equivalent globally consistent network in polynomial time– (After applying 2 & 3 consistency)

(Properties)

Nov

10 200929

Tractable Networks: LCC&SUC

• Arc consistency removes values from domains– Show: After removing any value v in Dy, still

LCC&SUC

• Case 1: any cxy, forest Tx on Dx, Ty on Dy

– Construct a new forest Ty’’ for y such that for every subchain of Tx, its image is still a subchain under Ty’’

– What we did in our last proof

(Proof)

Nov

10 200930

Tractable Networks: LCC&SUC• Case 2: any cyx, forest Tx on Dx , Ty on Dy

– If it is LCC&SUC we are done– Exists subchain ty of Ty such that it

contains v and image is no longer connected graph after v removed

– Let tx be image of ty before v removed• After v removed, breaks tx into two

chains• Let gap in tx be tr

– Let r be root and l last node in tr

• pv and pr be parents of v and l• cv and cl be children of v and l

– Consider any node u in tr

• u supported by v but not pv or cv in ty

• Since cxy LCC&SUC, image of tx must be a subchain containing (pv,v,cv)– image of u must be on

or contain subchain (pv,v,cv)

• v only support of u• u should also be

removed• after removal of tr, image

of ty is now connected and u is a subchain

pv

cv

v

pr

cl

r

u

l

ty

tx

Ty Tx

tr

(Proof)

Nov

10 200931

Tractable Networks: LCC&SUC• Path consistency preserves

LCC&SUO– cxz = cxz∩cyz◦cxy

• First: Composition of cxy and cyz is LCC&SUC– Any subchain tx in Dx, its image t’y under cxy is a subchain

– Since image of t’y with respect to cyz is a subchain of Dz

– Image of tx under composition is subchain of Dz

a

b

c d

a

b

c d

a

b

c d

x y z

cyzcxy

(Proof)

Nov

10 200932

Tractable Networks: LCC&SUC

• Second: show intersection is LCC&SUC, where

c’xz cyz◦cxy

c’’xz cxz c’xz– Subchain tx with only one value

of Dx

– Its images under cxz and c’xz are subchains of forest on Dz

– Intersection is still a subchain, so v’s image under c’’xz is subchain

a

b

c d

a

b

c d

a

b

c

d

x y z

cxy c’xzcxz= Dx Dz

cyz

(Proof)

Nov

10 200933

Tractable Networks: LCC&SUC– Subchain tx with more than

one value of Dx

– If image is subchain of Dz, done– Since intersection does not

form cycle• image of tx not connected

– Starting from root of tx

• Find first value (v) whose image is disjoint from image of parent

• Let a be last value of parents image

• Let d be root of v images• Let u be any value between a

and d

(Proof)x z

pv

v

c1 c2

a

c

d

b

Nov

10 200934

Tractable Networks: LCC&SUC– Show there is no support for u– pv images under

• cxz = I(pv), c’xz = I’(pv)

– I(pv)∩I’(pv) is subchain of Dz

– I(pv) and I’(pv) are chains• a is last value of one I(pv) or I’(pv)

– pv not in u’s image under czx

• I(u) has to be below parent since I(u) is chain

– I(v) and I’(v) images of v under cxz and c’xz• I(v) should include d and all values between a

and d in forest Dz

– Because cxz is LCC&SUC

• Since d is root of ∩, I’(v) includes d but not does include anything above d

– v is not support of u• I’(u) has to be above v

– Image of u under c’’xz is empty

x

pv

v

c1 c2

pv

v

c1 c2

z

a

c

d

b

a

c

d

b

c’xz

cxz

(Proof)

Nov

10 200935

Tractable Networks: LCC&SUC• Original constraint network

– Might not be constraint between x and y– Assume graph of original network is connected– So there must be a path– All constraints on path are LCC&SUC are closed– C’xy composition of constraints over path

• C’xy is also LCC&SUO

• Before enforcing path consistency set constraint between x y to be c’xy and repeat for any two variables without direct constraint

• Now before path consistency– Any two variables there is a constraint on them that is LCC&SUC

(Proof)

Nov

10 200936

Application• Scene Labeling (As seen

before from “On the Minimality and Global Consistency of Row-Convex Constraint Networks”)– Line with label

• Convex (+)• Concave (-)• Boundary (>)

– Junction constraints• Fork• Arrow• Ell

Forka b c d e

Arrowu v w

Ell1 2 3 4 5 6

+ +

+

- -

- -

- -

+ +-

- -+ +

++

--

Nov

10 200937

Application

b

c d e a

2

1 3

45

6

uv

w

Forka b c d e

Arrowu v w

Ell1 2 3 4 5 6

+ +

+

- -

- -

- -

+ +-

- -+ +

++

--

a b c d e

u 0 1 1 0 0

v 1 0 0 0 0

w 1 0 0 0 0

a b c d e

u 0 1 1 0 0

v 1 0 0 0 0

w 1 0 0 0 0a b c d e

u 0 1 1 0 0

v 1 0 0 0 0

w 1 0 0 0 0

1 2 3 4 5 6

u 0 0 1 0 0 0

v 0 0 0 0 0 1

w 0 0 0 0 1 1

1 2 3 4 5 6

u 1 0 0 0 0 0

v 0 0 0 1 0 0

w 0 0 0 0 1 1

3 7

1 5

2 6

4

c21=

c31=

c51= c24= c37= c56= c26= c34= c57=

Nov

10 200938

Related Work (1)• Jeavons and colleagues

– Characterize complexity of constraint languages• A constraint language over D is a set of relations with finite arity• CSP associated with language L, denoted CSP(L), are triple (V,D,C)

– V = arbitrary set of variables– D = domain of each variable of V– C = set of constraints

• Constraint Language L over D is tractable if CSP(L’) can be solved in poly• Set of problems (V,D,C)

– V = {1,2,…,n} – D = {D1,D2,…,Dn} Di = arbitrary finite set– C = set of constraints

• Tractability of constraint language is not the same as a set of problems– Constraint Language involves fixed domain and fixed set of relations– All variables in different instances of CSP(L) have same domain

Nov

10 200939

Related Work (2)

• Multi-sorted constraint languages over more than one set

• CSP associated with multi-sorted language over {D1,D2,…,Dk}– Variables in CSP(L) can take any Di

• Big gap between multi-sorted language and set of problems

Nov

10 200940

Related Work (3)

• Focus on constraint language L over D– Every relation R in L satisfies LCC&SUC– Enforcing arc and path consistency guarantees

global consistency– In this situation, [Jeavons et al. AIJ98] gives

general characterization of all constraint languages which enforcing k-consistency ensures global consistency• LCC&SUC explains specific subclcass of these languages

Nov

10 200941

Related Work (4)

• [Kumar 06] uses randomized algorithms to show the tractability of “arc consistent consecutive tree convex” (ACCTC) networks– No deterministic algorithm is known to exist for this purpose

• Efficient recognition of constraints– Known: tree convexity [Zhang & Bao 08]

– Still open • ACCTC• Connected row convexity• Locally chain convexity, and • Strictly union closedness

Nov

10 200942

Conclusion

Property Composition Intersection

Tree Convex Not closed ClosedConsecutiveness Closed Not closedLocal Chain Convexity Not closed ClosedLCC&SUC Closed Closed

LCC&SUC (PC global consistency)Question: How does LCC&SUC relate to row convexity (RW)? LCC&SUC is likely stronger than RW but authors do not show that RW is a special case of LCC&SUC

Nov

10 200943

Thank You

• Any questions?– I know I would…