Nov Properties of Tree Convex Constraints Authors: Yuanlin Zhang & Eugene C. Freuder Presentation by Robert J. Woodward CSCE990 ACP, Fall

Nov 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 2

Nov 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 3

Nov Introduction Binary constraint network Tree Convex – Construct a tree for a variables domain – All allowed supports must form subtree abcd a1110 b1011 c1101 a bcd {a,b, c,d} {a,b,c} xy c xy c xy = Tree for y 4 Linear-time algorithm can detect tree convexity [Conitzer+ AAAI04]

Nov Introduction Binary constraint network Tree Convex – Construct a tree for a variables domain – All allowed supports must form subtree bcd a110 b011 c101 b cd {b,c,d}{a,b,c} xy c xy c xy = Tree for y 5 Linear-time algorithm can detect tree convexity [Conitzer+ AAAI04]

Nov Definitions: Basic Constraint networks (Binary) – Variables: V = {x 1,x 2,…,x n } – Domains: D i for each x i ϵ V. Finite – Constraints: Between ordered variables Constraint between (x,y) is c xy c yx is a different constraint – Operations (on constraints) Intersection (∩) Composition (◦) Inverse 6

Nov Definitions: Support, Image For a constraint c xy Value u ϵ D x Value v ϵ D y – support: u and v satisfy c xy – image of u under c xy denoted I y (u) Set of supports in D y – image of a subset of D x Union of images of its values abcd a1110 b1011 c1101 {a,b, c,d} {a,b,c} xy c xy c xy = What is I y (b) ? { a, c, d } 7

Nov 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 8

Nov 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 9

Nov 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 10 a bd ef x a bc ∩ x yza bc Common Tree: d

Nov 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 11 a bd ef x a bc ∩ x yza bc Common Tree: d

Nov Definitions: Tree Convex Definition 1 – Sets E 1,…,E k are tree convex with respect to a forest T on U iϵ1..k E i – If every E i is a subtree of T Sets that are tree convex {a,b,c} {a,b,d} {a,d} {a,c} {d} … 12 a bcd

Nov Definitions: Tree Convexity on c xy Definition 2 – A constraint c xy is tree convex with respect to a forest T on D y if the images of all values of D x are tree convex with respect to T 13 abcd a1110 b1011 c1101 a bcd {a,b, c,d} {a,b,c} xy c xy c xy = Tree for y

Nov 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 14

Nov 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, E 1 and E 2 – Consistency lemma if network is path consistent I x (E 1 )∩I x (E 2 ) Ø – Since all constraints in l are tree convex Extension sets are tree convex – Tree convex sets intersection lemma ∩I x (E iϵL ) Ø iff I x (E j )∩I x (E k ) Ø – From consistency lemma, we have k -consistent Network is k+1 consistent iff any instantiation of k distinct variables and a new variable, ∩I c (E iϵL ) Ø Therefore, by induction – globally consistent 15 [Zhang &Yap 03]

Nov 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 16

Nov a b cd Properties: Intersection & Composition Assume c 1 xy c 2 xy are tree convex to a forest T on domain D y Prove there intersection is tree convex – Let c xy = c 1 xy  c 2 xy – For any v in D x images under c 1 xy and c 2 xy are subtrees of T intersection of two images is a subtree of T by Proposition 1 – Every image of every v in D x is a subtree of T – Therefore, c xy is tree convex Composition does not preserve tree convexity a b cd xy c 2 xy x y c 1 xy c 2 xy 17

Nov Properties: Consecutiveness Consecutive – Tree convex constraint c xy with respect to a forest T y on D y is consecutive with respect to a forest T x on D x – iff every two neighboring values a,b on T x, I y (a) U I y (b) is subtree of T y Constraint Network tree convex and consecutive – exists a forest on each domain – every constraint c xy is tree convex and consecutive with respect to the forests on D y and D x abcd a1110 b1010 c1101 a bcd {a,b, c,d} {a,b,c} xy c xy c xy = Tree for y a bc Tree for x {a,b,c} {a,b,c,d} 18

Nov Properties: Consecutiveness composition Class of consecutive tree convex constraints is closed under composition – Let c xy and c yz be two consecutive tree convex constraints with trees T x, T y, and T z – c xz is their composition – c xz is tree convex v in D x image under c xz = U bϵI y (v) I z (b) Union of images of neighboring values in I y (v) is subtree of T z, union of all values in I y (v) is a subtree too – c xz is consecutive u,v in D x and neighbors under T x Since c xy is consecutive, I y (u) U I y (v) is subtree to T y I y (u) U I y (v) is also a subtree of T z because of consecutiveness of C yz So, I z (u) U I z (v) is a subtree of T z 19 a b cd x a b cd y a b cd z c xy c yz

Nov Properties: Consecutiveness composition Class of consecutive tree convex constraints is closed under composition – Let c xy and c yz be two consecutive tree convex constraints with trees T x, T y, and T z – c xz is their composition – c xz is tree convex v in D x image under c xz = U bϵI y (v) I z (b) Union of images of neighboring values in I y (v) is subtree of T z, union of all values in I y (v) is a subtree too – c xz is consecutive u,v in D x and neighbors under T x Since c xy is consecutive, I y (u) U I y (v) is subtree to T y I y (u) U I y (v) is also a subtree of T z because of consecutiveness of C yz So, I z (u) U I z (v) is a subtree of T z 20 a b cd x a b cd y a b cd z c xy c yz

Nov 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 21

Nov 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 c xy with respect to a forest on D y – iff image of every value in D x is subchain of forest Constraint network locally chain convex – iff exists forest on each domain such that every constraint is locally chain convex abcd a1110 b1011 c1101 a bcd {a,b, c,d} {a,b,c} xy c xy c xy = Tree for y 22

Nov 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 D y is T y – Value v is removed from D y – Need to show every c xy in C is locally chain convex Could have made images on D x not connected Construct new forest T y ’’ on D y – broken subchains will be reconnected 23

Nov Tractable Networks: Locally Chain Convex Let v 1,…,v l be children of v Let p v be parent of v Construct a new forest T y ’ from T y Remove v and all edges incident on v Construct T y ’’ from T y ’ – Add edge between p v and all v i If v is root of T y, let T y ’’ be T y ’ 24 pvpv v v1v1 v2v2 pvpv v v1v1 v2v2 v1v1 v2v2 pvpv Ty’Ty’ T y ’’ TyTy

Nov Tractable Networks: Locally Chain Convex Let v 1,…,v l be children of v Let p v be parent of v Construct a new forest T y ’ from T y Remove v and all edges incident on v Construct T y ’’ from T y ’ – Add edge between p v and all v i If v is root of T y, let T y ’’ be T y ’ 25 Ty’Ty’ T y ’’ v v1v1 v2v2 TyTy v v1v1 v2v2 v1v1 v2v2

Nov 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 b cd a b cd a b cd xyz c yz c xy a b cd a b cd xz c xz 26 (Composition)

Nov Tractable Networks: LCC&SUC Locally chain convex and strictly union closed (LCC&SUC) – With respect to forest T x on D x and T y on D y – image of any subchain in T x is subchain in T y Constraint network is locally chain convex and strictly union closed – every constraint c xy is locally chain convex and strictly union closed with respect to the forests D x and D y a b cd a b cd a b cd xyz c yz c xy 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 27

Nov 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) 28 (Properties)

Nov Tractable Networks: LCC&SUC Arc consistency removes values from domains – Show: After removing any value v in D y, still LCC&SUC Case 1: any c xy, forest T x on D x, T y on D y – Construct a new forest T y ’’ for y such that for every subchain of T x, its image is still a subchain under T y ’’ – What we did in our last proof 29 (Proof)

Nov Tractable Networks: LCC&SUC Case 2: any c yx, forest T x on D x, T y on D y – If it is LCC&SUC we are done – Exists subchain t y of T y such that it contains v and image is no longer connected graph after v removed – Let t x be image of t y before v removed After v removed, breaks t x into two chains Let gap in t x be t r – Let r be root and l last node in t r p v and p r be parents of v and l c v and c l be children of v and l – Consider any node u in t r u supported by v but not p v or c v in t y Since c xy LCC&SUC, image of t x must be a subchain containing (p v,v,c v ) – image of u must be on or contain subchain (p v,v,c v ) v only support of u u should also be removed after removal of t r, image of t y is now connected and u is a subchain 30 pvpv cvcv v prpr clcl r u l tyty txtx TyTy TxTx trtr (Proof)

Nov Tractable Networks: LCC&SUC Path consistency preserves LCC&SUO – c xz = c xz ∩c yz ◦c xy First: Composition of c xy and c yz is LCC&SUC – Any subchain t x in D x, its image t ’ y under c xy is a subchain – Since image of t ’ y with respect to c yz is a subchain of D z – Image of t x under composition is subchain of D z 31 a b cd a b cd a b cd xyz c yz c xy (Proof)

Nov Tractable Networks: LCC&SUC Second: show intersection is LCC&SUC, where c ’ xz  c yz ◦c xy c ’’ xz  c xz c ’ xz – Subchain t x with only one value of D x – Its images under c xz and c ’ xz are subchains of forest on D z – Intersection is still a subchain, so v ’s image under c ’’ xz is subchain 32 a b cd a b cd a b c d xyz c xy c ’ xz c xz = D x  D z c yz (Proof)

Nov Tractable Networks: LCC&SUC – Subchain t x with more than one value of D x – If image is subchain of D z, done – Since intersection does not form cycle image of t x not connected – Starting from root of t x 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 33 (Proof) xz pvpv v c1c1 c2c2 a c d b

Nov Tractable Networks: LCC&SUC – Show there is no support for u – p v images under c xz = I(p v ), c ’ xz = I ’ (p v ) – I(p v )∩I’(p v ) is subchain of D z – I(p v ) and I ’ (p v ) are chains a is last value of one I(p v ) or I ’ (p v ) – p v not in u ’s image under c zx I(u) has to be below parent since I(u) is chain – I(v) and I ’ (v) images of v under c xz and c ’ xz I(v) should include d and all values between a and d in forest D z – Because c xz 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 34 x pvpv v c1c1 c2c2 pvpv v c1c1 c2c2 z a c d b a c d b c ’ xz c xz (Proof)

Nov 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 35 (Proof)

Nov 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 Fork abcde Arrow uvw Ell

Nov Application b cdea u v w Fork abcde Arrow uvw Ell abcde u01100 v10000 w10000 abcde u01100 v10000 w10000 abcde u01100 v10000 w u v w u v w c 21 = c 31 = c 51 =c 24 = c 37 = c 56 =c 26 = c 34 = c 57 =

Nov 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 = { D 1,D 2,…,D n } D i = 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 38

Nov Related Work (2) Multi-sorted constraint languages over more than one set CSP associated with multi-sorted language over { D 1,D 2,…,D k } – Variables in CSP( L ) can take any D i Big gap between multi-sorted language and set of problems 39

Nov 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 40

Nov 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 41

Nov Conclusion PropertyCompositionIntersection Tree ConvexNot closedClosed ConsecutivenessClosedNot closed Local Chain ConvexityNot closedClosed LCC&SUCClosed 42 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 Thank You Any questions? – I know I would… 43