Advanced Consistency Techniques

Transcription

1 Advanced Consistency Techniques Christophe Lecoutre CRIL CNRS UMR 8188 Universite d Artois Lens, France ACP Summer School

2 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 2

3 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 3

4 Bibliography Foundations of constraint satisfaction (Tsang) [Tsa93] Principles of Constraint Programming (Apt) [Apt03] Constraint Processing (Dechter) [Dec03] Constraint-based local search (van Hentenryck, Michel) [vhm05] Handbook of Constraint Programming [RvBW06] Constraint Propagation [Bes06] Constraint Networks (Lecoutre) [Lec09] Ant Colony Optimization and Constraint Programming (Solnon) [Sol10] 4

5 Notations Variables: x,y,z,... Values: a,b,c,... Domain of variable x: dom(x) Initial domain of variable x: dom init (x) Constraints: c,c xy,... Scope of constraint c: scp(c) Relation of constraint c: rel(c) Initial relation of constraint c: rel init (c) Tuples: τ,τ Constraint Networks (CN or CSP): P, P,... Set of variables of P: vars(p) Set of constraints of P: cons(p) 5

6 Illustration of CNs: the 8-Queens Problem ¾ ½ 6

7 Instances of Queens Problem as Constraint Networks For the n-queens instance, we have a constraint network P such that: vars(p) = {x a,...,x h } with dom(x i ) = {1,...,n}, i a..h cons(p) = {x i x j i j x i x j : i a..h, j a..h, i j} x a ¾ For the 3-queens instance, the compatibility graph is: ½ ½ ½ ¾ x b ¾ x c 7

8 A Solution to the 8-queens Instance ¾ ½ 8

9 Important Notions for Filtering Algorithms Let c be a r-ary constraint. Definition (Valid Tuples) A r-tuple τ is valid on c iff: x scp(c), τ[x] dom(x) The set of valid tuples on c is val(c) = Π x scp(c) dom(x). Definition (Supports = Tuples both Valid and Allowed) A r-tuple τ is a support on c iff: τ is a valid tuple on c which is allowed by c The set of supports on c is sup(c) = rel(c) val(c). 9

10 My constraint: pay 50 euros In my pocket: Pay with 5 Pay with 2 is allowed but not valid is valid but not allowed Pay with is allowed and valid 10

11 My constraint: c xy : x < y We have: dom init (x) = {0,1,2,3} and dom(x) = {1,2,3} dom init (y) = {0,1,2,3,4,5} and dom(y) = {0,1,2,3} dom init (x) dom init (y) y x ¼ ¼ ½ ¾ val(c xy ) ½ ¾ rel(c xy ) sup(c xy ) 11

12 My constraint: c xy : x < y We have: dom init (x) = {0,1,2,3} and dom(x) = {1,2} dom init (y) = {0,1,2,3,4,5} and dom(y) = {3,4} dom init (x) dom init (y) y x ¼ ¼ ½ ¾ ½ ¾ rel(c xy ) Question What is the state of the constraint? sup(c xy ) = val(c xy ) 12

13 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 13

14 Instantiation Definition (Instantiation) An instantiation I of a set X = {x 1,...,x k } of variables is a set {(x 1,a 1 ),..., (x k,a k )} such that: a1 dom init (x 1 ),..., a k dom init (x k ). We denote X by vars(i ). An instantiation I is valid iff (x,a) I, a dom(x). An instantiation I on a CN P is an instantiation of a set X vars(p) ; it is complete if vars(i ) = vars(p), partial otherwise. 14

15 Locally Consistent Instantiation Definition Let I be an instantiation. I covers a constraint c iff scp(c) vars(i ). I satisfies a constraint c iff 1 I covers c 2 the tuple (a 1,...,a r ), such that I [scp(c)] = {(x 1,a 1 ),...,(x r,a r )}, is allowed by c, i.e. (a 1,...,a r ) rel(c). Definition (Locally Consistent Instantiation) An instantiation I on a constraint network P is locally consistent iff 1 I is valid 2 every constraint of P that is covered by I is satisfied by I Otherwise, it is said to be locally inconsistent. 15

16 Example 1/6 x ¾ ¼ z ½ ½ ¼ ¾ ¼ ½ ¾ y 16

17 Example 2/6 x ¾ ¼ z ½ ½ ¼ ¾ ¼ ½ ¾ y {(x,1)} is a locally consistent instantiation as there is no constraint covered by it. 17

18 Example 3/6 x ¾ ¼ z ½ ½ ¼ ¾ ¼ ½ ¾ y {(x,1),(y,0)} is not a locally consistent instantiation since the constraint c xy covered by it is not satisfied by it. 18

19 Example 4/6 x ¾ ¼ z ½ ½ ¼ ¾ ¼ ½ ¾ y {(x,1),(y,1)} is a locally consistent instantiation since the constraint c xy covered by it is satisfied by it. 19

20 Example 5/6 x ¾ ¼ z ½ ½ ¼ ¾ ¼ ½ ¾ y {(x,1),(y,1),(z,0)} is a complete but not locally consistent instantiation since the constraint c xz is not satisfied by it. 20

21 Example 6/6 x ¾ ¼ z ½ ½ ¼ ¾ ¼ ½ ¾ y {(x,1),(y,1),(z,2)} is a complete and locally consistent instantiation since all variables are covered and all constraints satisfied. Question What is the connection with the basic backtracking search algorithm? 21

22 Solution Definition (Solution) A solution of a CN P is a complete instantiation on P that is locally consistent. The set of solutions of P is denoted by sols(p). 22

23 The Constraint Network Note that a CSP instance is trivially unsatisfiable when a domain or a relation is empty. We shall consider all such instances as equivalent, and shall denote by the representative of this implicit equivalence class. Notation Let P be a constraint network. We shall write P = iff x vars(p) dom(x) = or c cons(p) rel(c) =. 23

24 Nogood Definition (Globally Inconsistent Instantiation) An instantiation I on a constraint network P is globally inconsistent iff it cannot be extended to a solution of P, globally consistent otherwise. Definition (Nogood) A globally inconsistent instantiation is also called a (standard) nogood. Remark A locally inconsistent instantiation is a nogood. The reverse is not necessarily true. 24

25 Illustration ¾ ½ {(x a,3),(x b,6),(x c,4),(x d,2),(x e,7)} is locally inconsistent this is a nogood. 25

26 Illustration ¾ ½ {(x a,3),(x b,1),(x c,4),(x d,2)} is globally inconsistent this is a nogood. 26

27 Interest of nogoods? Use it! To make the CN more explicit. 27

28 Nogood Representation The nogood representation x of a variable x is the set of instantiations: { {(x,a)} a dom init (x) \ dom(x) } The nogood representation c of a constraint c, with scp(c) = {x 1,...,x r }, is the set of instantiations: {(x 1,a 1 ),...,(x r,a r )} (a 1,...,a r ) dom init (x) \ rel(c) x scp(c) The nogood representation P of a constraint network P is the set of instantiations: x c x vars(p) c cons(p) 28

29 Example P c xy a 1 a 2 a 3 x = {{(x, a 1 )}} x ỹ = {{(y, b 2 )}, {(y, b 3 )}} b 1 b 2 b 3 y c xy = {{(x, a 1 ), (y, b 3 )}, {(x, a 2 ), (y, b 3 )}, {(x, a 3 ), (y, b 1 )}, {(x, a 3 ), (y, b 2 )}} P = {{(x, a 1 )}, {(y, b 2 )}, {(y, b 3 )}, {(x, a 1 ), (y, b 3 )}, {(x, a 2 ), (y, b 3 )}, {(x, a 3 ), (y, b 1 )}, {(x, a 3 ), (y, b 2 )}} 29

30 General Partial Ordering Definition Let P and P be two CNS such that vars(p) = vars(p ). P P iff P P. P P iff P P. For example, P P : when P is obtained from P by removing some values from variable domains (we also use d ), or when P is obtained from P by removing some tuples from constraint relations. 30

31 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 31

32 Consistencies A consistency is a property defined on CNs. Typically, it reveals some nogoods. We shall only consider nogood-identifying consistencies. A 1-order consistency (or domain-filtering consistency) allows to identify inconsistent values (nogoods of size 1). For example: Generalized Arc Consistency (GAC), Path Inverse Consistency (PIC) Singleton Arc Consistency (SAC) A 2-order consistency allows to identify inconsistent pairs of values (nogoods of size 2). For example: Path Consistency (PC), Dual Consistency (DC), Conservative variants of PC and DC 32

33 The most famous Consistency: GAC Definition (Generalized Arc Consistency) A constraint c is GAC-consistent iff x scp(c), a dom(x), there exists a support for (x,a) on c. A constraint network P is GAC-consistent iff every constraint of P is GAC-consistent. At a finer level, we have: A value (x,a) of P is GAC-consistent iff for every constraint c of P involving x, there exists a support for (x,a) on c. Remark GAC is referred to as Arc Consistency (AC) when constraints are binary. GAC is also called hyper arc-consistency or domain consistency. 33

34 Example ½ ½ ¾ ¾ ½ ¾ ½ Figure: A Sudoku Grid. 34

35 Example dom(x) = {2, 5} w x ½ dom(z) = {2, 5, 8, 9} dom(w) = {3, 5, 7} y z dom(y) = {2, 5, 9} Figure: An alldifferent constraint involving nine variables, of which w, x, y and z are unfixed. The other five variables are fixed. This constraint is not generalized arc-consistent. The values (w,3), (w,5) and (z,8) have no support on this constraint (and are said to be GAC-inconsistent). 35

36 Consistency φ Definition (φ-consistent Network) Let φ be a consistency. A constraint network P is said to be φ-consistent iff the property φ holds on P. Most of the consistencies mentioned in the literature are well-behaved. This means that for any CN P, there exists a greatest φ-consistent network, denoted by φ(p), smaller than (or equal to) P while preserving the set of solutions. Theorem (Monotonicity) Let P and P be two CNs and φ be a well-behaved consistency. P P φ(p) φ(p ) 36

37 Pre-order on Consistencies Definition Let φ and ψ be two consistencies. φ is stronger than (or equivalent to) ψ iff whenever φ holds on a constraint network P, ψ also holds on P. φ is strictly stronger than ψ iff φ is stronger than ψ and there exists at least one constraint network P such that ψ holds on P but not φ. Theorem Let φ and ψ be two well-behaved consistencies. φ is stronger than ψ iff for every CN P, we have φ(p) ψ(p). 37

38 Domain-filtering Consistencies (partial view) È ÖÆÁ Ë ÆÁ ÁÛ ÈÏ ÅÜÊÈ ÈÁ ÅÜÊÈÏ ÖÈÁ Ë φ ψ ÑÒ φ ØÖØÐÝ ØÖÓÒÖ ØÒ ψ ÒÖÝ ÆØÛÓÖ ÆÓÒ¹ÒÖÝ ÆØÛÓÖ 38

39 Defining φ-consistent values To completely define a domain-filtering consistency φ, it is sufficient to define the conditions under which a value (x,a) of a CN P is considered as φ-consistent. A value (x,a) of P is GAC-consistent iff for any constraint c of P involving x, there exists a support for (x,a) on c. A value (x,a) of a P is PIC-consistent iff for every set {y,z} of two variables of P, with x y and x z, there exists b dom(y) and c dom(z) such that {(x,a),(y,b),(z,c)} is locally consistent. A value (x,a) of P is SAC-consistent iff GAC(P x=a )

40 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 40

41 Enforcing GAC on Table Constraints Many schemes/algorithms proposed in the literature: GAC-valid GAC-allowed [BR97] GAC-valid+allowed [LS06] NextIn Indexing [LR05] NextDiff Indexing [GJMN07] Using Tries [GJMN07] Compressed Tables [KW07] Using MDDs [CY10] STR [Ull07, Lec08] Our focus 41

42 A Table Constraint as a MDD table[c xyz ] x y z ½ µ mdd(c xyz ) ¾ µ µ µ µ µ µ µ µ ÐÚÐ ½ ¾ x y z Ø ½¼ µ (a) A table (b) A MDD 42

43 Algorithm 1: enforcegac-mdd(c: MDD constraint): set of variables Output: the set of variables in scp(c) with reduced domain Σ true Σ false foreach variable x scp(c) do gacvalues[x] exploremdd(mdd(c)) // gacvalues is updated during exploration // domains are now updated and X evt computed X evt foreach variable x scp(c) do if gacvalues[x] dom(x) then dom(x) gacvalues[x] X evt X evt {x} return X evt 43

44 Algorithm 2: exploremdd(node: Node): Boolean Output: true iff node is supported if node = t then return true if node Σ true then return true if node Σ false then return false // since we are at a leaf // since already proved to be supported // since already proved to be unsupported x node.variable ; supported false foreach arc node.outs do if arc.value dom(x) then if exploremdd(arc.destination) then supported true gacvalues[x] gacvalues[x] {arc.value} if supported = true then Σ true Σ true {node} else Σ false Σ false {node} return supported 44

45 Simple Tabular Reduction Simple tabular reduction (STR) is an original approach introduced by J. Ullmann. The principle of STR is to dynamically maintain the tables (of supports). To summarize, GAC is enforced while removing invalid tuples, and consequently, only supports are kept in tables. Efficency is obtained by using a sparse set data structure. 45

46 Algorithm 3: STR(c: constraint): set of variables Output: the set of variables in scp(c) with reduced domain foreach variable x scp(c) do gacvalues[x] foreach tuple τ table[c] do if isvalid(c,τ) then foreach variable x scp(c) do if τ[x] / gacvalues[x] then add τ[x] to gacvalues[x] else removetuple(c,τ) // domains are now updated and X evt computed // as in Algorithm

47 Illustration with STR table[c xyz ] x y z µ µ µ µ µ µ µ 47

48 Illustration with STR table[c xyz ] x y z dom(y) dom(x) dom(z) µ µ µ µ µ µ µ 47

49 Illustration with STR table[c xyz ] x y z dom(y) dom(x) dom(z) µ µ µ µ µ µ µ gacv alues[x] = {} gacv alues[y] = {} gacv alues[z] = {} 47

50 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a} gacv alues[y] = {a} gacv alues[z] = {c} 47

51 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a} gacv alues[y] = {a} gacv alues[z] = {c} 47

52 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a} gacv alues[y] = {a, c} gacv alues[z] = {b, c} 47

53 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a} gacv alues[y] = {a, c} gacv alues[z] = {b, c} 47

54 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a} gacv alues[y] = {a, c} gacv alues[z] = {b, c} 47

55 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a, c} gacv alues[y] = {a, c} gacv alues[z] = {b, c} 47

56 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a, c} gacv alues[y] = {a, c} gacv alues[z] = {b, c} 47

57 Illustration with STR dom(z) dom(y) dom(x) table[c xyz ] x y z µ µ µ µ µ µ µ gacv alues[x] = {a, c} gacv alues[y] = {a, c} gacv alues[z] = {b, c} 47

58 ÙÖÖÒØ ØÐ Maintaining Current Tables position[c xyz ] table[c xyz ] x y z currentlimit[c xyz ] levellimits[c xyz ] ¾ ½ ¼ ½¼... ¹½ ¹½ ¹½ ½ ¾ ½¼ ½ ¾ ½¼ ½ ¾ ½¼ µ µ µ µ µ µ µ µ µ µ Figure: Initialization of STR data structures for a ternary positive table constraint c xyz. 48

59 ÙÖÖÒØ ØÐ Maintaining Current Tables position[c xyz ] table[c xyz ] x y z currentlimit[c xyz ] levellimits[c xyz ] ¾ ½ ¼... ¹½ ½¼ ¹½ ½ ¾ ½¼ ½ ¾ ½¼ ½ ¾ ½¼ µ µ µ µ µ µ µ µ µ µ Figure: STR applied after the removal of (y,b) at level 1. (z,c) no longer has support and will therefore be deleted. 49

60 ÙÖÖÒØ ØÐ Maintaining Current Tables position[c xyz ] table[c xyz ] x y z currentlimit[c xyz ] levellimits[c xyz ] ¾ ½ ¼... ½¼ ¹½ ½ ¾ ½¼ ½ ¾ ½¼ ½ ¾ ½¼ µ µ µ µ µ µ µ µ µ µ Figure: STR applied after the removal of (y,c) at level 2. No value will be deleted. 50

61 ÙÖÖÒØ ØÐ Maintaining Current Tables position[c xyz ] table[c xyz ] x y z currentlimit[c xyz ] levellimits[c xyz ] ¾ ½ ¼... ¹½ ½¼ ¹½ ½ ¾ ½¼ ½ ¾ ½¼ ½ ¾ ½¼ µ µ µ µ µ µ µ µ µ µ Figure: Structures obtained after the restoration performed at level 1. 51

62 ÙÖÖÒØ ØÐ Maintaining Current Tables position[c xyz ] table[c xyz ] x y z currentlimit[c xyz ] levellimits[c xyz ] ¾ ½ ¼ ½¼... ¹½ ¹½ ¹½ ½ ¾ ½¼ ½ ¾ ½¼ ½ ¾ ½¼ µ µ µ µ µ µ µ µ µ µ Figure: Structures obtained after the restoration performed at level 0. 52

63 Experiments on Crossword Puzzles Class. GAC schemes STR Series val all va str str2 str2+ words-vg5-5 cpu #nodes=38 mem 4.9M 4.9M 4.8M 4.7M 4.8M words-vg5-6 cpu #nodes=718 mem 6.5M 6.5M 6.3M 6.2M 6.3M words-vg5-7 cpu #nodes=6,957 mem 8.4M 8.4M 8.2M 8.1M 8.2M words-vg5-8 cpu #nodes=256k mem 4.6M 10M 10M 10M Crossword puzzles with dictionary words (45,371 words). Time-out of 1,200 seconds set per instance. 53

64 Experiments on Crossword Puzzles Class. GAC schemes STR Series val all va str str2 str2+ uk-vg5-5 cpu #nodes=28 mem 12M 12M 12M 12M 12M uk-vg5-6 cpu #nodes=145 mem 17M 17M 16M 16M 16M uk-vg5-7 cpu #nodes=408 mem 22M 22M 22M 22M 22M uk-vg5-8 cpu #nodes=8,148 mem 12M 12M 11M 11M 11M Crossword puzzles with dictionary uk (225,349 words). Time-out of 1,200 seconds set per instance. 54

65 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 55

66 DC Definition (Dual Consistency - DC) Let P be a constraint network. A locally consistent instantiation {(x,a),(y,b)} on P is DC-consistent iff (y,b) GAC(P x=a ) and (x,a) GAC(P y=b ). P is DC-consistent iff every locally consistent instantiation {(x,a),(y,b)} on P is DC-consistent. 56

67 CDC Definition (Conservative Dual Consistency - CDC) Let P be a constraint network. A locally consistent instantiation {(x,a),(y,b)} on P is CDC-consistent iff either c cons(p) scp(c) = {x,y} or {(x,a),(y,b)} is DC-consistent. P is CDC-consistent iff every locally consistent instantiation {(x,a),(y,b)} on P is CDC-consistent. 57

68 Properties Proposition DC is strictly stronger than PC On binary CNS, DC is equivalent to PC Proposition For any constraint network P, we have: But GAC DC(P) = sdc(p) GAC CDC(P) = scdc(p) AC CPC(P) scpc(p) AC PPC(P) sppc(p). sφ is φ + (G)AC 58

69 Relationships between 2-order Consistencies È ÈÈ ÈÈ È φ ψ ÑÒ φ ØÖØÐÝ ØÖÓÒÖ ØÒ ψ È È 59

70 scdc1 Algorithm 4: scdc1 P GAC(P) ; x first(vars(p)) marker x repeat if dom(x) > 1 then if revise-scdc 1(x) then P GAC(P) ; marker x x nextcircular(x,vars(p)) until x = marker // GAC is initially enforced // GAC is maintained 60

71 scdc1 Algorithm 5: revise-scdc1(var x: variable): Boolean modified false foreach value a dom P (x) do P GAC(P x=a ) ; // Singleton check on (x,a) if P = then remove a from dom P (x) ; // SAC-inconsistent value modified true else foreach constraint c xy cons(p) do foreach value b dom P (y) do if b / dom P (y) then remove (a,b) from rel P (c xy ) ; /* CDC-inconsistent pair of values */ modified true return modified 61

72 Example v w x y z

73 Example v w x y z with wiped-out domain

74 Example v w x y z

75 Example v w x y z

76 Example v w x y z

77 Example v w x y z

78 Example v w x y z

79 Example v w x y z

80 Results spc8 scdc1 sdc2 400 CPU (in seconds) tightness t (in %) Figure: Mean results obtained with PC algorithms on classes of 100 random binary instances of class 50,90,1225,t. 63

81 Impact for Search Instance MAC scdc1-mac scen11-f8 CPU nodes 14,068 4,946 scen11-f6 CPU nodes 302K 145K scen11-f4 CPU nodes 2,826K 1,834K scen11-f3 CPU 2,338 1,725 nodes 12M 5,863K scen11-f2 CPU 7,521 5,872 nodes 37M 21M scen11-f1 CPU 17,409 13,136 nodes 93M 55M 64

82 Impact for Search For the test instance B8x8With2Hints, MAC : 49 runs, 125 seconds scdc1+mac : 45 runs, 72 seconds. 65

83 Outline 1 Introduction 2 Instantiations and Orderings 3 Consistencies 4 Efficient GAC Algorithms for Table Constraints 5 Dual Consistency 6 Consistencies based on Failed Values 66

84 Lemma The following lemma is directly derived from [Freuder and Hubbe, 1993] Lemma If a value (x,a) of a CN P is globally inconsistent then every solution S of P is such that S[x/a] violates at least one constraint of P involving x. If P is a binary CN, then every solution of P necessarily contains a value for a variable y x which is not compatible with (x,a). 67

85 Illustration w x C x I x C y I y C z y I z z Solid edges represent allowed tuples Dashed edges represent forbidden tuples 68

86 Failed Values and Conflict Sets Intuitively, a failed value is a value removed from a CN because it has been proved to be (globally) inconsistent. Definition Let P be a CN, x be a variable of P and a dom init (x). The conflict set of (x,a) on a constraint c of P involving x, denoted by χ(c,x,a), is the set of instantiations I of scp(c) \ {x} on P such that I {(x,a)} does not satisfy c. The conflict set of (x,a) on P is χ(x,a) = c cons(p) x scp(c) χ(c,x,a). 69

87 Conflict Sets a b c x a b c w y a b c a b c z χ(w, a) = {} χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

88 Conflict Sets a b c x a b c w y a b c a b c z χ(w, a) = {{(x, b)}, χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

89 Conflict Sets a b c x a b c w y a b c a b c z χ(w, a) = {{(x, b)}, {(y, a), (z, a)}} χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

90 Conflict Sets a b c x a b c w y a b c a b c z χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

91 Conflict Sets a b c x a b c w y a b c a b c z χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

92 Conflict Sets a b c x a b c w y a b c a b c z χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

93 Conflict Sets a b c x a b c w y a b c a b c z χ(w, a) = {{(x, b)}, {(y, a), (z, a)}} χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 70

94 FVC Definition Let (x,a) be a failed value of a CN P and I an instantiation on P. (x,a) is covered by I iff vars(χ(x,a)) vars(i ). (x,a) is verified by I iff J χ(x,a) J I. Definition ([LR09]) Let P be a constraint network. An instantiation I on P is FVC-consistent for a failed value (x,a) of P iff either (x,a) is not covered by I or (x,a) is verified by I. An instantiation I on P is FVC-consistent iff it is FVC-consistent for every failed value of P; otherwise, I is said to be FVC-inconsistent. 71

95 Illustration of FVC a b c x a b c w y a b c a b c z (w, c) Ð ÚÐÙ ÙÑÔØÓÒµ χ(w, c) = {{(x, b)}, {(x, c)}, {(y, c), (z, c)}} 72

96 Illustration of FVC a b c x a b c w y a b c a b c z I = {(x, a), (y, c), (z, c)} ÚÖ (w, c) Ù I ÓÒØÒ {(y, c), (z, c)} χ(w, c) 72

97 Illustration of FVC a b c x a b c w y a b c a b c z I = {(x, a)} ιÓÒ ØÒØ Ù (w, c) ÒÓØ ÓÚÖ Ý I 72

98 Illustration of FVC a b c x a b c w y a b c a b c z I = {(x, a), (y, a), (z, a)} ιÒÓÒ ØÒØ Ù (w, c) ÓÚÖ Ò ÒÓØ ÚÖ Ý I 72

99 Proposition Proposition Any FVC-inconsistent instantiation is globally inconsistent. Otherwise stated, some nogoods can be identified via deleted values (that are themselves nogoods). These nogoods are not necessarily of size 1. 73

100 Illustration w x C x I x C y I y C z y I z z Every tuple in C x C y C z is a nogood (of size 3) 74

101 AFVC Definition Let P be a constraint network. A value (x,a) of P is AFVC-consistent for a failed value (y,b) of P iff (x,a) can be extended to a locally consistent instantiation verifying (y,b). A value (x,a) of P is AFVC-consistent iff (x,a) is AFVC-consistent for every failed value of P; otherwise, (x,a) is said to be AFVC-inconsistent. P is AFVC-consistent iff every value of P is AFVC-consistent. 75

102 Proposition Proposition Any AFVC-inconsistent value is globally inconsistent. Note that: AFVC is a domain-filtering consistency AFVC can be regarded as a local consistency since it suffices to reason from the conflict set of each failed value. In particular for binary constraints, a value (x,a) is AFVC-consistent for a failed value (y,b) iff (x,a) is compatible with a valid value in χ(y,b). 76

103 Illustration of AFVC w a b... x a b c z a b a c y (w, a) Ð ÚÐÙ ÙÑÔØÓÒµ χ(w, a) = {{(x, a)}, {(y, c)}} 77

104 Illustration of AFVC w a b... x a b c z a b a c y (w, b) ιÒÓÒ ØÒØ Ò (w, b) ÒÓÑÔØÐ ÛØ ÚÐÙ Ò χ(w, a) 77

105 Illustration of AFVC w a b... x a b c a b z... a c y (z, a) ιÒÓÒ ØÒØ Ò (z, a) ÒÓÑÔØÐ ÛØ ÚÐÙ Ò χ(w, a) 77

106 Hierarchy of consistencies ÅÜÎ ÈÁÎ ËÎ Î ØÖØÐÝ ØÖÓÒÖ ÒÓÑÔÖÐ 78

107 Preliminary Results (binary instances) MAC MAC+FVC MAC+AFVC Instance CPU nodes CPU nodes CPU nodes Graph coloring 1-fullins M , ,636 2-fullins , , ,764 2-insertions , , ,753 2-insertions , , ,941 Composed composed M M M333 composed M , ,636 Job-shop os-taillard M , ,845 os-taillard , , ,818 os-taillard , , ,226 Queen Attacking qa , , ,940 qa M M M576 79

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