Properties of Tree Convex Constraints 1,2

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1 Properties of Tree Convex Constrints 1,2 Yunlin Zhng, Eugene C Freuder b Deprtment of Computer Science, Texs Tech University, Lubbock, USA b Cork Constrint Computtion Center, University College Cork, Irelnd Abstrct It is known tht tree convex network is globlly consistent if it is pth consistent However, if tree convex network is not pth consistent, enforcing pth consistency on it my not mke it globlly consistent In this pper, we investigte the properties of some tree convex constrints under intersection nd composition As result, we identify sub-clss of tree convex networks tht re loclly chin convex nd strictly union closed This clss of problems cn be mde globlly consistent by rc nd pth consistency nd thus is trctble Interestingly, we lso find tht some scene lbeling problem cn be modeled by tree convex constrints in nturl nd meningful wy Key words: Constrint networks, tree convex constrints, row convex constrints, connected row convex constrints, scene lbeling problem 1 Introduction A binry constrint network is tree convex [20] if we cn construct tree for the domin of ech vrible so tht for ny constrint, no mtter wht vlue one vrible tkes, ll the vlues llowed for the other vrible form subtree of the constructed tree As n exmple, the constrint c xy of Fig 1() is tree convex while c xy of Fig 1(c) is not Corresponding uthor Emil ddresses: yzhng@csttuedu (Yunlin Zhng), efreuder@4cuccie (Eugene C Freuder) 1 A preliminry version of this pper ppered in [18] 2 This work hs received support from Science Foundtion Irelnd under Grnt 00/PI1/C075 Preprint submitted to Elsevier Science 27 July 2007

2 b c d c xy = b c b c d c xy = () (b) (c) Fig 1 () Constrint c xy is represented by mtrix The column {, b, c} beside the mtrix is the domin of x, nd the row {, b, c, d} bove the mtrix is the domin of y (b) A tree constructed for the vlues of the domin of y c xy is tree convex with respect to this tree (c) c xy is obtined from c xy by deleting the vlue from the domin of y c xy is not tree convex with respect to ny tree Tree convex constrints further the study of the convexity nd monotonicity of constrints [14,13] It hs been shown tht tree convex network is globlly consistent if it is pth consistent However, if tree convex network is not pth consistent, enforcing pth consistency on it my not mke it globlly consistent becuse some constrints my be modified during the enforcing procedure nd thus my no longer be tree convex In this pper, we exmine the tree convex constrints nd chrcterize conditions under which the desirble tree convex property of network is preserved when rc nd pth consistency re enforced We then identify trctble clss of restricted tree convex constrints This result generlizes the erlier work on monotone [13] nd connected row convex constrints [5] The ltter is built on the work of [14] Finlly, we show tht tree convex constrints help to model some scene lbeling problems in nturl nd meningful wy The relted work by Jevons et l [7,8] nd Kumr [10] will be discussed in the lst section 2 Preliminries In this section, we review the bsic concepts nd nottions used in this pper Constrint Networks A binry constrint network consists of set of vribles V = {x 1, x 2,, x n } with finite domin D i for ech vrible x i V, nd set of binry constrints C over the vribles of V c xy denotes constrint on vribles x nd y which is defined s reltion over D x nd D y Opertions on reltions, eg, intersection ( ), composition ( ), nd inverse, re pplicble to constrints We ssume tht, between ny ordered vribles (x, y), there is only one constrint c xy nd c yx re considered to be two different constrints However, we 2

3 ssume the inverse of c xy is equl to c yx Imge Given constrint c xy nd vlue u D x, v D y is support of u if u nd v stisfy c xy, tht is (u, v) c xy The imge of u under c xy, denoted by I y (u), is the set of ll its supports in D y The imge of subset of D x is the union of the imges of its vlues k-consistency A constrint network is k-consistent if ny consistent instntition of ny distinct k 1 vribles cn be consistently extended to ny new vrible A network is strongly k-consistent if it is j-consistent for ll j k A strongly n-consistent network is clled globlly consistent 2- nd 3-consistency re usully clled rc consistency nd pth consistency respectively Note tht, under this definition, we need to dd universl constrint between vribles tht re not explicitly constrined by the network More mterils on these concepts cn be found in [11,13,6] Forests, Trees, Chins, nd Sets In the following we review trees tht ply fundmentl role in the nlysis of tree convex constrints nd introduce some new nottions used in this pper A forest is grph without ny cycles A tree is connected grph without ny cycles A forest cn be regrded s set of trees In the rest of the pper, we lwys ssume there is root for tree in forest The pth between ny two nodes (or vertices) of tree is unique nd the distnce of node to the root is defined s the number of edges in the pth between them Given tree, subtree is defined s connected subgrph of the tree, nd its root is the node closest to the root of the tree A forest on set S is forest whose vertex set is exctly S We lso cll set I subtree of forest T if there exists subtree of some tree in T such tht its vertex set is exctly I An empty set is subtree of ny forest A tree (nd subtree respectively) becomes chin (nd subchin respectively) if ech of its nodes hs t most one child The lst vlue (or node) of subchin is the frthest one wy from its root For exmple, the grph in Fig 1(b) is tree on {, b, c, d} {, b, c} is subtree of it, nd {, b} is subchin whose lst vlue is b The intersection of two trees is defined s the grph whose vertices nd edges re in both trees It hs the following property: Proposition 1 [20] Let T 1, T 2 be two subtrees of some tree The intersection of T 1 nd T 2 is lso subtree of the tree Furthermore, if the intersection is not empty, the root of the intersection is either the root of T 1 or tht of T 2 Next, we relx the tree structures, used in some concepts in [20], to the forest structures 3

4 Definition 1 Sets E 1,, E k re tree convex with respect to forest T on E i if every E i is subtree of T i 1l For exmple, given the tree in Fig 1(b), sets {, b, c}, {, b, d}, nd {, c, d} re tree convex Definition 2 A constrint c xy is tree convex with respect to forest T on D y if the imges of ll vlues in D x re tree convex with respect to T Exmple Consider c xy in Fig 1() The imges of, b, c re {, b, c}, {, c, d}, nd {, b, d} respectively They re tree convex with respect to the tree in Fig 1(b) nd thus c xy is tree convex with respect to tht tree The reders re invited to verify tht there is no tree to mke c xy (in Fig 1(c)) tree convex In [20], tree convex constrint network is defined s network where ll constrints re tree convex with respect to common tree on the union of ll domins in the network In the following definition, only the forests on the individul domins mtter Definition 3 A constrint network is tree convex if there exists forest on the domin of ech vrible such tht every constrint c xy of the network is tree convex with respect to the forest on D y As pointed out by one of the referees, the new definition of tree convexity of constrint networks is equivlent to the old ones [20] if the domins of the vribles re disjoint Given ny problem, we cn mke the domins of the vribles disjoint by renming the vlues of the domins of the vribles so tht they re different from those of the domins of the other vribles The renming preserves the solutions of constrint network One dvntge of the new definition is tht even if the domins of two vribles shre some vlues, it explicitly llows us to construct different forests for them in deciding the tree convexity of the network More importntly, in this pper, we need to introduce further restrictions (eg, consecutiveness) on tree convex constrints The forest-bsed definition helps to simplify the presenttion, the proofs, nd the understnding of the results The tree convex set intersection lemm in [20] still holds for the new definition of tree convex sets, which cn be lifted to the following consistency result Proposition 2 A tree convex constrint network is globlly consistent if it is pth consistent The proof follows directly from tht of [20] becuse the new definition does not ffect the essentil prt of tht proof 4

5 3 Properties of Intersection nd Composition of Tree Convex Constrints A network cn be mde pth consistent by removing from the constrints the tuples which cn not be consistently extended to new vrible It is equivlent to the mtrix computtion c xy = c xy (c zy c xz ), where denotes composition To mke use of Theorem 2, we need to study the impct of the intersection nd composition opertions on the tree convexity of constrints Intersection preserves tree convexity Proposition 3 Assume constrints c 1 xy nd c 2 xy re tree convex with respect to forest T on the domin D y Their intersection is lso tree convex Proof Let c xy = c 1 xy c 2 xy For ny v D x, its imges under c 1 xy nd c 2 xy re both subtrees of T The intersection of the two imges is subtree of T by Proposition 1 Tht is, the imge of every v D x is subtree of T Hence, c xy is tree convex However, the composition of tree convex constrints might not preserve the tree convexity Let us use more intuitive wy thn mtrix multipliction to understnd the composition Consider the constrints in Fig 2 After composing c xy nd c yz, the imge of under the composition c xz is {, b, c, d} tht is exctly the union of the imges of b nd d in D y under c yz To ssure tht the imge of under c xz is tree, we cn simply require tht I z (b) I z (d) is (sub)tree, tht is I z (b) nd I z (d) touch ech other x c xy y c yz z x c xz z b b b c d c d c d Fig 2 The composition of two constrints In the digrms of this pper, vlue is drwn s dot or letter, nd vrible is drwn s n ellipse The vlues inside n ellipse form the domin of the corresponding vrible The edges between two ellipses specify the constrint between the corresponding vribles Definition 4 A tree convex constrint c xy with respect to forest T y on D y is consecutive with respect to forest T x on D x if nd only if for every two neighboring vlues, b on T x, I y () I y (b) is subtree of T y A constrint network is tree convex nd consecutive iff there exists forest on ech domin such tht every constrint c xy is tree convex nd consecutive with respect to the forests on D y nd D x Proposition 4 The clss of consecutive tree convex constrints is closed under composition 5

6 Proof Let c xy nd c yz be two consecutive tree convex constrints with respect to forests T x, T y nd T z on D x, D y nd D z respectively, nd c xz the composition of c xy nd c yz Firstly, we show tht c xz is tree convex Consider ny v D x Let its imge in D y be I y (v) The imge of v under c xz would be b Iy(v)I z (b) where I z (b) is the imge of b under c yz Since the union of the imges of ny neighboring vlues in I y (v) is subtree of T z, the union of ll the imges of vlues of I y (v) is subtree of T z Secondly, we show tht c xz is consecutive Let u, v D x be neighbors under T x Let I z (u) nd I z (v) be their imges under c xz Since c xy is consecutive, I y (u) I y (v) is subtree of T y Hence, the union of the imges (with respect to c yz ) of the vlues of I y (u) I y (v) is subtree of T z due to the consecutiveness of c yz Therefore, I z (u) I z (v) is subtree of T z 4 Trctble Tree Convex Constrint Networks The intersection of two subtrees my be n empty set, which mens tht, fter the intersection of two tree convex constrints, the imge of vlue could be empty Deleting such vlue could mke constrint no longer tree convex, which is shown by the exmple in Fig 1 It is lso interesting to note tht constrint c xy my become tree convex fter sufficient number of vlues re removed from D y The following specil clss of tree convex constrints tht is closed under the opertion of deleting vlues Definition 5 A constrint c xy is loclly chin convex with respect to forest on D y if nd only if the imge of every vlue in D x is subchin of the forest A constrint network is loclly chin convex iff there exists forest on ech domin such tht every constrint c xy is loclly chin convex with respect to the forest on D y For exmple, under the tree for D y in Fig 1(b), the constrint in Fig 1() is not loclly chin convex becuse the imge of D x is {, b, c} tht is not subchin of the tree on D y In fct, there does not exist ny tree to mke it chin convex Proposition 5 A loclly chin convex constrint network (V, D, C) is still loclly chin convex fter the removl of ny vlue from ny domin Proof Assume the forest on D y is T y nd vlue v is removed from D y The removl of v does not ffect the property of ny constrint c yx C We need to show tht every c xy C is loclly chin convex The deletion of v could 6

7 mke the imges of some vlues of D x not connected By constructing new forest T y on D y, those broken subchins would be connected under T y Let the children of v be v 1,, v l nd the prent of v be p v Construct new forest T y from T y by removing v nd ll edges incident on v If v is the root of some tree of T y, let T y be T y The imge of ny vlue of D x either contins v or not In the ltter cse, the imge is still chin In the former cse, v is the shllowest node of the imge, chin, nd thus the imge is still chin fter the removl of v If v is not the root of ny tree of T y, construct T y from T y by dding n edge between p v nd v i for ll i(1 i l) The imge of ny vlue of D x is subchin of T y To identify trctble clss of tree convex constrints, first ttempt is to combine the locl chin convexity (for deleting vlue) with consecutiveness (for composition) However, the composition my destroy the chin convexity, s shown by the exmple in Fig 3() c x b c xy y b c d c d d b () c yz z y c yx x p v p r v r l c v c l (b) Fig 3 In this digrm, we drw the tree on domin inside n ellipse () Both c xy nd c yz re loclly chin convex, but their composition is not becuse the imge of b D x under this composition is {b, c, d} (the drkened shpe) tht is not subchin (b) t y contins the solid lines in D y t x contins (p r, r, l, c l ) t r contins (r, l) The imge of vlue under the composition is the union of severl subchins This union cn not be gurnteed to be subchin by the consecutiveness of the constrints We need stronger restriction Definition 6 A constrint c xy is loclly chin convex nd strictly union closed with respect to forests T x on D x nd T y on D y iff the imge of ny subchin of T x is subchin of T y Remrk Locl chin convexity nd strict union closedness imply consecutiveness of constrint network; but consecutiveness of loclly chin convex network might not imply strict union closedness s shown by the exmple in Fig 3 Now we introduce the clss of constrint networks tht is closed under the removl of vlue, nd the intersection nd composition of constrints Definition 7 A constrint network is loclly chin convex nd strictly union closed iff there exists forest on ech domin such tht every constrint c xy 7

8 of the network is loclly chin convex nd strictly union closed with respect to the forests on D x nd D y Theorem 1 A loclly chin convex nd strictly union closed constrint network (V, D, C) cn be trnsformed to n equivlent globlly consistent network in polynomil time Proof We show tht the given network is loclly chin convex fter rc nd pth consistency re enforced on it In ccordnce with Theorem 2, the new network is globlly consistent It is known tht rc nd pth consistency enforcing [19] re of polynomil complexity Since rc consistency enforcing only removes vlues from domins, we show tht fter the removl of ny vlue v D y the network is still loclly chin convex nd strictly union closed Tht is, we show tht ll constrints c xy, c yx C re loclly chin convex nd strictly union closed Cse 1 Consider ny c xy C nd the forests T x on D x nd T y on D y Similr to the proof of Proposition 5, we cn construct new forest T y for y such tht for every subchin of T x, its imge is still subchin under T y Cse 2 Consider ny constrint c yx C nd the forests T x on D x nd T y on D y If it is still loclly chin convex nd strictly union closed, we re done Otherwise, there exists subchin t y of T y such tht it contins v nd its imge is no longer connected grph due to the removl of v See Fig 3(b) Let t x be the imge of t y before removing v After the removl of v, t x is broken into two chins Let the gp (removed subchin) in t x be t r Note t r might not be equl to the imge of v due to the possible overlpping of the imge of v nd tht of its prent nd/or child Let r be the root nd l the lst node of t r Let p v nd p r be the prents of v nd r respectively, nd c v nd c l the children of v nd l respectively Consider ny node u t r We know tht u is supported by v, but not by p v or by c v in t y Further, since c xy is loclly chin convex nd strictly union closed, the imge of t x must be subchin contining (p v, v, c v ) It implies tht the imge of u must be on or contin the subchin (p v, v, c v ) Hence, v is the only support of u After v is gone, u should lso be removed After the removl of t r, the imge of t y is now connected nd thus subchin Next, we show tht pth consistency enforcing preserves the locl chin convexity nd strict union closedness For ny constrint c xz, pth consistency is usully done by first composing c xy nd c yz, nd then setting the new constrint between x nd z to be the intersection of c xz nd c yz c xy Firstly, we show tht the composition of c xy nd c yz is loclly chin convex nd strictly union closed Assume c xy nd c yz re loclly chin convex nd strictly union closed with respect to the forests T x, T y nd T z on D x, D y nd D z respectively For ny subchin t x D x, its imge t y under c xy is subchin 8

9 Since the imge of t y with respect to c yz is subchin of D z, the imge of t x under the composition is subchin of D z c xz c xz x z p v v b c d () c xz x p v v (b) z b c d p v x v c xz (c) b c d z Fig 4 () c xz c xz In the intersection, ssume b nd c re not shred by the imges of v under c xz nd under c xz The constrints c xz nd c xz should hve form s shown in (b) nd (c) Secondly, we show tht the intersection, c xz, of c xz nd c xz (= c xy c yz ) is loclly chin convex nd strictly union closed Consider subchin, with only one vlue, of D x Its imges under c xz nd c xz re subchins of the forest on D z Their intersection is still chin nd thus v s imge under c xz is subchin Consider subchin t x, with more thn one vlues, of D x In this prgrph, when we refer to n imge, it is under c xz If the imge of t x is subchin of D z, we re done Otherwise, let t z be the imge of t x t z is not subchin Since the intersection does not form cycle, t z must not be connected Strting from the root of t x, we find the first vlue v t x whose imge is disjoint from the imge of its prent p v Assume the imge of v is below tht of p v (the opposite cn be proved similrly) Let be the lst vlue of p v s imge Let d be the root of v s imge See Fig 4() Let u be ny vlue between (but not including) nd d in D z We next prove tht there is no support for u Hence, vlues between nd d should be removed nd the imge of t x is chin fter the deletion Let p v s imges under c xz nd c xz be I(p v ) nd I (p v ) respectively The intersection of I(p v ) nd I (p v ) is subchin of D z Since both I(p v ) nd I (p v ) re chins, must be the lst vlue of either I(p v ) or I (p v ) Assume it is the lst vlue of I(p v ) See Fig 4(b) It implies p v is not in u s imge I(u) under c zx, since u is between nd d I(u) hs to be below p v (not including it) becuse I(u) is chin Let I(v) nd I (v) be the imges of v under c xz nd c xz respectively I(v) should include t lest d nd ll vlues between nd d in the tree D z becuse c xz is loclly chin convex nd strictly union closed Since d is the root of I(v) I (v), I (v) includes d but does not include vlues bove d (see Fig 4(c)) Hence, v is not support of u (under c xz), implying tht I (u) hs to be bove v (not including it) Therefore, the imge of u under c xz is empty becuse it is the intersection of I(u) nd I (u) In other words, u hs no support in the intersection of c xz nd c xz 9

10 Now we re ble to discuss cse ignored in the previous discussion In the originl constrint network, there might not be ny constrint between some vribles, sy x nd y Without loss of generlity, we ssume the grph of the originl network is connected Therefore, there must be pth from x to y All constrints on the pth re loclly chin convex nd strictly union closed By the result in the previous prgrphs, the intersection nd composition of loclly convex nd strictly union closed constrints re closed Let c xy be the composition of the constrints over the pth in order The constrint c xy is loclly chin convex nd strictly union closed Now, before enforcing pth consistency (nd possibly one more round of rc consistency), set the constrint between x nd y to be c xy nd repet this for ny two vribles without direct constrint on them After this modifiction, for ny two vribles there is constrint on them tht is loclly chin convex nd strictly union closed Hence, the constrint network is loclly chin convex fter enforcing pth consistency nd thus is globlly consistent 5 An Appliction of Tree Convex Networks In this section, we exmine the ppliction of tree convex constrints to scene lbeling problem Given two dimensionl line drwing of physicl world of plne-fced objects, the scene lbeling problem is to identify from the drwing the physicl objects nd their sptil reltions with the requirement tht the identifiction grees with humn being Wltz nd others reduce this problem into problem of ssociting line with lbel such tht set of concrete constrints on the junctions re stisfied Given line drwing, junction is defined s the mximum set of lines tht intersect t the sme point Note tht lines of drwing correspond to edges of physicl objects, nd junctions correspond to vertexes of physicl objects There re only three types of edges tht line cn represent: convex, concve, nd boundry edges tht re denoted by the lbels +, - nd > respectively An edge is the intersection of two surfces of n object It is convex if it cn be touched by bll from the front For exmple, when there is cube in front of viewer, its top edge is convex for the viewer An edge is concve if it cn never be touched by bll For exmple, when viewer fces wll nd floor, their intersection edge is concve becuse there is no wy to mke bll touch the edge from the front A boundry edge is the intersection of the bckground nd surfce of n object of concern An excellent exposition of scene lbeling problems cn be found in the book [16], nd detiled tretment of this topic cn be found in [15] Scene lbeling problems re NP-hrd [9] In the following, we show tht some scene lbeling instnce cn be modeled nturlly by tree convex constrints 10

11 nd solved efficiently Consider the line drwing in Fig 5 tken from [14] This drwing involves three types of junctions: Fork, Arrow, nd Ell The shpe of junction 1 is Fork, tht of the junctions 3 nd 5 is n Arrow, nd tht of junctions 4, 6 nd 7 is n Ell To lbel this drwing is to find solution of constrint network defined s follows We introduce vrible x i for ech junction i A vlue for vrible is wy to lbel the lines in the corresponding junction Under pproprite ssumptions, there re only 5 physiclly relizble wys to lbel Fork, 3 n Arrow, nd 6 n Ell, which re listed in Fig 5 The constrints on the vribles re strightforwrd, ie, ny two vribles should tke the sme lbel on their shred line All the constrints re listed s mtrices in Fig A line drwing Fork Arrow u v w Ell b c d e Fig 5 The left is line drwing, nd the right is tble of the lbelings for vrious junctions The letter bove ech lbeling of junction is its nme by which the lbeling is referred to in the rest of this section A distinctive feture of this model is tht the vlues of vrible hve complex structures nd there is some nturl reltionship mong them Consider the vlues for Fork junction in Fig 5 Vlues c, d, nd e hve n edge lbeled s -, nd ll three edges of b re lbeled s - We cn let b be the prent of c, d nd e, resulting in the subtree {b, c, d, e} in Fig 6() Since vlue hs nothing to do with the rest, it forms tree itself Similrly, we hve the forests for Arrow vlues in Fig 6(b) nd Ell vlues in Fig 6(c) Under these forests, the constrints re loclly chin convex nd strongly union closed For exmple, consider the constrint c 21 on vribles x 2 nd x 1 in Fig 6 The domin of x 1 is shown in Fig 6 (), nd tht of x 2 in Fig 6(b) It cn be verified tht the imge of every subchin of the forest of x 2 is subchin of the forest of x 1 Note tht n empty set is tken s (trivil) subchin of ny tree By Theorem 1, this network is globlly consistent fter rc nd pth consistency re enforced on it In this exmple, we hve identified the forest structures for the domins in n intuitive nd meningful wy A more generl lesson is tht by studying the semntics of domin vlues, we could discover more efficient constrint solving techniques 11

12 bcde u c 21 = v c 31 = c 51 = w c b d e () Fork v u w (b) Arrow u c 24 = c 37 = c 56 = v w c 26 = c 34 = c 57 = (c) Ell Fig 6 The constrints for lbeling the drwing in Fig 5 6 Relted Work nd Conclusion Jevons nd collegues hve done series of work to chrcterize the complexity of constrint lnguges [3] A constrint lnguge is prmeterized by set Given set D, constrint lnguge over D is set of reltions with finite rity Given lnguge L over D, the constrint stisfction problems ssocited with L, denoted by CSP(L), re triple (V, D, C) where V is n rbitrry set of vribles, D (over which L is defined) the domin of ech vrible of V, nd C the set of constrints over the vribles such tht ech c C belongs to L A constrint lnguge L over D is trctble if CSP(L ) cn be solved in polynomil time, for ech finite subset L L Severl types of polymorphism hve been identified to chrcterize the trctble lnguges In this pper, insted of constrint lnguge, we consider the trctbility of set of problems (V, D, C) where V = {1, 2,, n}, D = {D 1, D 2,, D n } nd D i (n rbitrry finite set) is the domin of vrible i (i 1n), nd C set of constrints Our result shows tht this set of problems cn be solved in polynomil time when certin convexity properties re stisfied Although the trctbility of constrint lnguge seems to be the sme s the trctbility of set of problems, they re indeed different The key difference lies in tht constrint lnguge involves fixed domin D nd fixed set of reltions L All vribles in different instnces of CSP(L) hve to hve the sme domin D A recent work [2] hs generlized constrint lnguges to multi-sorted constrint lnguges tht re over more thn one set For multi-sorted constrint lnguge L over {D 1, D 2,, D k }, the vribles in CSP(L) re llowed to tke ny D i (i 1k) s their domins It is shown tht even this simple extension hs serious consequences for the chrcteriztion of constrint lnguges: [the originl constrint lnguges] cn in fct msk the difference between trctbility nd NP-completeness for some lnguges, [2] Not ll results for constrint lnguges hold for multi-sorted lnguges There is still gp between multi-sorted lnguge nd set of problems In 12

13 the CSPs ssocited with multi-sorted lnguge, the domins of vribles re restricted to fixed collection of sets while in set of problems, rbitrry set is llowed to be the domin of vrible The knowledge is still bsent on how lgebric opertions cn be used to directly chrcterize the trctbility of set of problems To hve better understnding of the reltionship between our result nd the results on constrint lnguges, we focus on constrint lnguges Consider constrint lnguge L over D Prticulrly, every reltion R L stisfies the convexity property mentioned in Theorem 1 Since enforcing rc nd pth consistency gurntees globl consistency of CSP(L) (by Theorem 1), L must hve ner-unnimity polymorphism by the result in [8] In this sitution, the result in [8] gives generl chrcteriztion ( indirectly through lgebric opertions) of ll constrint lnguges on which enforcing locl (k-) consistency ensures globl consistency while our result helps to identify concrete subclss of these lnguges ( directly through the convexity properties of the constrints) Bsed on the work reported here, Kumr [10] hs proposed more generl property on tree convexity rc consistent consecutive tree convexity (AC- CTC) such tht the problems with tht property re trctble Rdiclly different from our nd Jevons nd collegues pproches, Kumr uses rndom lgorithms s tool to show the trctbility of the problems of concern Due to the nture of our pproch, enforcing rc nd pth consistency on our proposed clss of problems ensures globl consistency, nd there is efficient deterministic lgorithms to chieve the globl consistency [12,1] For the ACCTC problems, it is not known whether there is efficient deterministic lgorithms, neither is it known whether the rc nd pth consistency ensures globl consistency The tree convexity of constrint network cn be recognized efficiently [17] Kumr lso observes tht n lgorithm in [4] cn be used to recognize the tree convexity of network lthough no lgorithm is presented to recognize the ACCTC of network As the cse of connected row convexity nd ACCTC, how to recognize efficiently whether constrint network is loclly chin convex nd strictly union closed is n open problem We hve presented some properties of tree convex constrints tht re closed under intersection nd/or composition As result, we identified new trctble clss of networks loclly chin convex nd strictly union closed networks on which enforcing rc nd pth consistency on them ensures globl consistency This result generlizes the existing work on convexity of constrints, eg, [5], nd revels more fundmentl property locl chin convexity nd strict union closedness tht determines the trctbility of clss of convex constrints Our result lso shows direct interction between the semntics of constrints nd the semntics of domin vlues in deciding trctble clss of problems This interction is reflected in the properties of intersection nd 13

14 composition of tree convex constrints An ppliction of the new trctble clss of networks is lso presented, demonstrting tht tree convexity is useful nd nturl wy to chrcterize the semntics of domin vlues, in ddition to the trditionl ones like totl ordering References [1] C Bessiere, JC Regin, RHC Yp, nd Y Zhng An optiml corse-grined rc consistency lgorithm Artificil Intelligence, 165(2): , 2005 [2] Andrei A Bultov nd Peter Jevons An lgebric pproch to multi-sorted constrints In CP, pges , 2003 [3] Dvid Cohen nd Peter Jevons The complexity of constrint lnguges In F Rossi, P vn Beek, nd T Wlsh, editors, Hndbook of Constrint Progrmming, pges Elsevier, 2006 [4] Vincent Conitzer, Jonthn Derryberry, nd Tuoms Sndholm Combintoril uctions with structured item grphs In AAAI, pges , 2004 [5] Y Deville, O Brette, nd P Vn Hentenryck Constrint stisfction over connected row convex constrints In Proceedings of Interntionl Joint Conference on Artificil Intelligence 1997, volume 1, pges , Ngoy, Jpn, 1997 IJCAI Inc (See lso Artificil Intelligence 109(1999): ) [6] EC Freuder Synthesizing constrint expressions Communictions of ACM, 21(11): , 1978 [7] P G Jevons, D A Cohen, nd M Gyssens Closure properties of constrints Journl of The ACM, 44(4): , 1997 [8] Peter Jevons, Dvid A Cohen, nd Mrtin C Cooper Constrints, consistency nd closure Artif Intell, 101(1-2): , 1998 [9] L M Kirousis nd C H Ppdimitriou The complexity of recognizing polyhedrl scenes In Journl of Computer nd System Sciences, volume 37, pges 14 38, 1988 [10] T K Stish Kumr Simple rndomized lgorithms for trctble row nd tree convex constrints In AAAI, 2006 [11] A K Mckworth Consistency in networks of reltions Artificil Intelligence, 8(1): , 1977 [12] R Mohr nd TC Henderson Arc nd pth consistency revisited Artificil Intelligence, 28: , 1986 [13] U Montnri Networks of constrints: fundmentl properties nd pplictions Informtion Science, 7(2):95 132,

15 [14] P vn Beek nd R Dechter On the minimlity nd globl consistency of row-convex constrint networks Journl of The ACM, 42(3): , 1995 [15] D L Wltz Generting semntic descriptions from drwings of scenes with shdows Technicl Report MAC-AI-TR-271, MIT, Cmbridge, MA, 1972 [16] Ptrick Henry Winston Artificil Intelligence Addison-Wesley, Reding, Msschusetts, third edition, 1992 [17] Guy Yosiphon Efficient lgorithm for identifying tree convex constrints Mnuscript, 2003 [18] Yunlin Zhng nd Eugene C Freuder Trctble tree convex constrints In Proceedings of Ntionl Conference on Artificil Intelligence 2004, pges , Sn Jose, CA, USA, 2004 AAAI press [19] Yunlin Zhng nd Rolnd H C Yp Mking AC-3 n optiml lgorithm In Proceedings of Interntionl Joint Conference on Artificil Intelligence 2001, pges , Settle, 2001 IJCAI Inc [20] Yunlin Zhng nd Rolnd H C Yp Consistency nd set intersection In Proceedings of Interntionl Joint Conference on Artificil Intelligence 2003, pges , Acpulco, Mexico, 2003 IJCAI Inc 15