1 / 43

Properties of Tree Convex Constraints

Properties of Tree Convex Constraints. Authors: Yuanlin Zhang & Eugene C. Freuder Presentation by Robert J. Woodward CSCE990 ACP, Fall 2009. Overview. Introduction Definitions Basic, support, image, consistency, trees, tree intersection, tree convex constraint, tree convex CSPs,

umeko
Download Presentation

Properties of Tree Convex Constraints

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Properties of Tree Convex Constraints Authors: Yuanlin Zhang & Eugene C. Freuder Presentation by Robert J. Woodward CSCE990 ACP, Fall 2009

  2. 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

  3. 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

  4. Introduction {a,b,c} {a,b, c,d} • Binary constraint network • Tree Convex • Construct a tree for a variables domain • All allowed supports must form subtree • Linear-time algorithm can detect tree convexity [Conitzer+ AAAI04] cxy x y a cxy= b c d Tree for y

  5. Introduction {a,b,c} {b,c,d} • Binary constraint network • Tree Convex • Construct a tree for a variables domain • All allowed supports must form subtree • Linear-time algorithm can detect tree convexity [Conitzer+ AAAI04] cxy x y b cxy = c d Tree for y

  6. Definitions: Basic • Constraint networks (Binary) • Variables: V = {x1,x2,…,xn} • Domains: Di for each xiϵ V. Finite • Constraints: Between ordered variables • Constraint between (x,y) is cxy • cyx is a different constraint • Operations (on constraints) • Intersection (∩) • Composition (◦) • Inverse

  7. Definitions: Support, Image {a,b,c} {a,b, c,d} • For a constraint cxy • Value u ϵDx • 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 cxy x y cxy= • What is Iy(b)? • {a,c,d}

  8. 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

  9. 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

  10. Definitions: Tree Intersection Common Tree: • 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 x y a z b c d e f a x ∩ b d a b c

  11. Definitions: Tree Intersection Common Tree: • 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 x y a z b c d e f a x ∩ b d a b c

  12. Definitions: Tree Convex • Definition 1 • Sets E1,…,Ekare tree convex with respect to a forest T on Uiϵ1..kEi • If every Ei is a subtree of T a • Sets that are tree convex • {a,b,c} • {a,b,d} • {a,d} • {a,c} • {d} • … b c d

  13. Definitions: Tree Convexity on cxy {a,b,c} {a,b, c,d} • 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 cxy x y a cxy = b c d Tree for y

  14. Definitions: Tree Convex CSP • 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

  15. Definition: Proof [Zhang &Yap 03] • 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)ØiffIx(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

  16. 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

  17. Properties: Intersection & Composition c1xy y x • Assume c1xy c2xyare tree convex to a forest T on domain Dy • Prove there intersection is tree convex • Let cxy = c1xy c2xy • For any v in Dx • images under c1xy and c2xy 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 a c2xy x y b b c c d d c1xy c2xy

  18. Properties: Consecutiveness {a,b,c} {a,b, c,d} • 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 cxy x y a {a,b,c} cxy = {a,b,c,d} b c d a b c Tree for y Tree for x

  19. Properties: Consecutiveness composition x y z • 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 Dxand 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 a a b b b c c c d d d cxy cyz

  20. Properties: Consecutiveness composition x y z • 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 Dxand 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 a a b b b c c c d d d cxy cyz

  21. 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

  22. Tractable Networks: Locally Chain Convex {a,b,c} {a,b, c,d} • 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 cxy x y a cxy = b c d Tree for y

  23. 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

  24. Tractable Networks: Locally Chain Convex • Let v1,…,vlbe 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 pv pv v v v2 v1 v2 v1 v2 v1 Ty Ty’ Ty’’

  25. Tractable Networks: Locally Chain Convex • Let v1,…,vlbe 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’ v v v2 v1 v2 v1 v2 v1 Ty Ty’ Ty’’

  26. Tractable Networks: Locally Chain Convex (Composition) x y z • 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 a a a a b b b b b c c c c c d d d d d cxy cyz x z cxz

  27. Tractable Networks: LCC&SUC x y z • 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 a a b b b c c c d d d cxy cyz • 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

  28. Tractable Networks: LCC&SUC (Properties) • 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)

  29. Tractable Networks: LCC&SUC (Proof) • 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

  30. Tractable Networks: LCC&SUC (Proof) • 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 • Case 2: any cyx, forest Tx on Dx , Ty on Dy • If it is LCC&SUC we are done • Exists subchainty 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 pr tx pv r u tr ty v l cv cl Ty Tx

  31. Tractable Networks: LCC&SUC (Proof) x y z • Path consistency preserves LCC&SUO • cxz = cxz∩cyz◦cxy • First: Composition of cxy and cyz is LCC&SUC • Any subchaintx 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 a a b b b c c c d d d cxy cyz

  32. Tractable Networks: LCC&SUC (Proof) x y z • Second: show intersection is LCC&SUC, where c’xz cyz◦cxy c’’xz cxzc’xz • Subchaintx 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’’xzis subchain a a a b b b c c c d d d cxy cyz c’xz cxz= Dx Dz

  33. Tractable Networks: LCC&SUC (Proof) x z • Subchaintx with more than one value of Dx • If image is subchain of Dz, done • Since intersection does not form cycle • image of txnot 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 pv a b v c c1 c2 d

  34. Tractable Networks: LCC&SUC (Proof) x z • 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 c’xz pv a b v c c1 c2 d cxz pv a b v c c1 c2 d

  35. Tractable Networks: LCC&SUC (Proof) • 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

  36. 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 + + - - - - + - - + + - - - + + + - + -

  37. 7 3 Application 1 5 4 2 6 b + + - - - - c d e a + - - c21= v u + + - - w - + + 2 4 5 c31= + - 1 3 6 + - c51= c24= c37= c56= c26= c34= c57=

  38. 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

  39. 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

  40. 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 subclcassof these languages

  41. 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

  42. Conclusion 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

  43. Thank You • Any questions? • I know I would…

More Related