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Foundations of Constraint Processing CSCE421/821, Fall 2004:

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## Foundations of Constraint Processing CSCE421/821, Fall 2004:

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**Consistency Algorithms**Foundations of Constraint Processing CSCE421/821, Fall 2004: www.cse.unl.edu/~choueiry/F04-421-821/ Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 123B choueiry@cse.unl.edu Tel: +1(402)472-5444 Consistency: Properties & Algorithms**Lecture Sources**Required reading • Algorithms for Constraint Satisfaction Problems, Mackworth and Freuder AIJ'85 • Sections 3.1, 3.2, 3.3. Chapter 3. Constraint Processing. Dechter Recommended • Sections 3.4—3.10. Chapter 3. Constraint Processing. Dechter • Networks of Constraints: Fundamental Properties and Application to Picture Processing, Montanari, Information Sciences 74 • Consistency in Networks of Relations, Mackworth AIJ'77 • Constraint Propagation with Interval Labels, Davis, AIJ'87 • Path Consistency on Triangulated Constraint Graphs, Bliek & Sam-Haroud IJCAI'99 Consistency: Properties & Algorithms**Outline**• Motivation and material • Node and arc consistency and their complexity • Criteria for performance comparison and CSP parameters. • Which AC algorithm to choose? Project results may tell! • Path consistency and its complexity • K-consistency, minimality and decomposability • Other related results: box consistency, higher levels of consistency, (i, j)-consistency, etc. Consistency: Properties & Algorithms**Consistency checking**Motivation • CSP are solved with search (conditioning) • Search performance is affected by: - problem size - amount of backtracking • Backtracking may yield Thrashing Exploring non-promising sub-trees and rediscovering the same inconsistencies over and over again Malady of backtrack search Consistency: Properties & Algorithms**V2**V1 {1, 2, 3} {1, 2, 3} V3<V1 V3<V2 V3 { 0, 1, 2} V3>0 V3<V4 V3<V5 V4 V5 {1, 2} { 1, 2} V5<V4 (Some) causes of thrashing Search order: V1, V2, V3, V4, V5 What happens in search if we: • do not check the constraint on V3 (node inconsistency) • do not check constraint between V3 and V5 (arc inconsistency) • do not check constraints between V3—V4, V3—V5, and V4—V5 (path inconsistency) Consistency: Properties & Algorithms**Consistency checking**Goal: eliminate inconsistent combinations Algorithms(for binary constraints): • Node consistency • Arc consistency (AC-1, AC-2, AC-3, ..., AC-7, AC-3.1, etc.) • Path consistency (PC-1, PC-2, DPC, PPC, etc.) • Constraints of arbitrary arity: • Waltz algorithm ancestor of AC's • Generalized arc-consistency (Dechter, Section 3.5.1 ) • Relational m-consistency (Dechter, Section 8.1) Consistency: Properties & Algorithms**Overview of Recommended Reading**• Davis, AIJ'77 • focuses on the Waltz algorithm • studies its performance and quiescence for given: • constraint types (order relations, bounded diff, algebraic, etc.) • domains types (continuous or finite) • Mackworth, AIJ'77 • presents NC, AC-1, AC-2, PC-1, PC-2 • Mackworth and Freuder, AIJ'85 • studies their complexity Mackworth and Freuder concentrate on finite domains More people work on finite domains than on continuous ones Continuous domains can be quite tough Consistency: Properties & Algorithms**Constraint Propagation with Interval Labels Ernest Davis**• `Old' paper (1987): terminology slightly different • Interval labels: continuous domains • Section 8: sign labels (discrete) • Concerned with Waltz algorithm (e.g., quiescence, completeness) • Constraint types vs. domain types (Table 3) • Addresses applications from reasoning • about Physical Systems (circuit, SPAM) • about time relations (TMM) • Advice: read Section 3, more if you wish Consistency: Properties & Algorithms**Waltz algorithm for label inference**• Refine(C(Vi, Vk, Vm, Vn), Vi) - finds a new label for Vi consistent with C. • Revise(C (Vi, Vk, Vm, Vn)) refines the domains of Vi, Vk, Vm, Vn by iterating over these variables until quiescence (no domains is further refined) • Waltz Algorithm revises all constraints by iterating over each constraint connected to a variable whose domain has been revised • Waltz Algorithm ancestor of arc-consistency Consistency: Properties & Algorithms**WARNING**→ Completeness: Running the Waltz Algorithm does not solve the problem. A=2 and B=3 is still not a solution! →Quiescence: The Waltz algorithm may go into infinite loops even if problem is solvable x [0, 100] x = y y [0, 100] x = 2y →Davis provides a classification of problems in terms of completeness and quiescence of the Waltz algorithm f(constraint types, domain types), see Table 3 Consistency: Properties & Algorithms**Importance of this paper** Establishes that constraints of bounded differences (temporal reasoning) can be efficiently solved by the Waltz algorithm (O(n3), n number of variables) Early paper that attracts attention on the difficulty of label propagation in interval labels (continuous domains). This work has been continued by Faltings (AIJ 92) who studied the early-quiescence problem of the Waltz algorithm. Consistency: Properties & Algorithms**Consistency checking algorithms**Examining finite, binary CSPs and their complexity Notation: Given variables i, j, k with values x, y, z, the predicate Pijk(x, y, z) is true iff the 3-tuple x, y, z Cijk Node consistency: checking Pi(x) Arc consistency: checking Pij(x,y) Path consistency: bit-matrix manipulation Consistency: Properties & Algorithms**time**space Worst-case asymptotic complexity Worst-case complexity as a function of the input parameters Upper bound: f(n) is O(g(n)) means that f(n)c.g(n) f grows as g or slower Lower bound:f(n) is (h(n)) means that f(n) c.h(n) f grows as g or faster Input parameters for a CSP: n = number of variables a = (max) size of a domain dk = degree of Vk ( n-1) e = number of edges (or constraints) [(n-1), n(n-1)/2] Consistency: Properties & Algorithms**Outline**• Motivation and material • Node & arc consistency and their complexity • Criteria for performance comparison and CSP parameters. • Which AC algorithm to choose? Project results may tell! • Path consistency and its complexity • K-consistency, minimality and decomposability • Other related results: box consistency, higher levels of consistency, (i, j)-consistency, etc. Consistency: Properties & Algorithms**Node consistency (NC)**Procedure NC(i): Di Di { x | Pi(x) } Begin for i 1 until n do NC(i) end Consistency: Properties & Algorithms**Complexity of NC**Procedure of NC(i) DiDi { x | Pi(x) } Begin for i 1 until n do NC(i) end • For each variable, we check a values • We have n variables, we do n.a checks • NC is O(a.n) Consistency: Properties & Algorithms**Vj**Vi Vi Vj 1 1 1 2 2 2 2 3 3 3 Arc-consistency Adapted from Dechter Definition: Given a constraint graph G, • A variable Vi is arc-consistent relative to Vj iff for every value aDVi, there exists a value bDVj | (a, b)CVi,Vj. • The constraint CVi,Vj is arc-consistent iff • Vi is arc-consistent relative to Vj and • Vj is arc-consistent relative to Vi. • A binary CSP is arc-consistent iff every constraint (or sub-graph of size 2) is arc-consistent Consistency: Properties & Algorithms**Procedure Revise**Revises the domains of a variable i Procedure Revise(i,j): Begin DELETE false for each x Dido if there is no y Dj such that Pij(x, y) then begin delete x from Di DELETE true end return DELETE end • Revise is directional • What is the complexity of Revise? {a2} Consistency: Properties & Algorithms**Revise: example R. Dechter**Apply the Revise procedure to the following example Consistency: Properties & Algorithms**Vj**Vi Vi Vj 1 1 1 2 2 2 2 3 3 3 Effect of Revise Adapted from Rina Dechter Question: Given • two variables Vi and Vj • their domains DVi and DVj, and • the constraint CVi,Vj, write the effect of the Revise procedure as a sequence of operations in relational algebra Hint: • Think about the domain DVi as a unary constraint CVi and • consider the composition of this unary constraint and the binary one.. • Solution: DVi DViVi (CVi,VjDVi) • This is actually equivalent to DViVi (CVi,VjDVi) Consistency: Properties & Algorithms**Arc consistency (AC-1)**Procedure AC-1: 1 begin 2 for i 1 untilndo NC(i) 3 Q {(i, j) | (i,j) arcs(G), i j } 4 repeat 5 begin 6 CHANGE false 7 for each (i, j) Q do CHANGE ( REVISE(i, j) or CHANGE ) 8 end 9 until ¬ CHANGE 10 end • AC-1 does not update Q, the queue of arcs • No algorithm can have time complexity below O(ea2) Consistency: Properties & Algorithms**V1**V1 V1 V1 V3 V2 V3 V3 V2 V3 V2 V2 V1 V1 V1 V3 V2 V3 V3 V2 V2 Arc consistency • AC may discover the solution Example borrowed from Dechter Consistency: Properties & Algorithms**Arc consistency**2. AC may discover inconsistency Example borrowed from Dechter Consistency: Properties & Algorithms**x**[0, 10] x [0, 7] x [4, 7] x y-3 x y-3 x y-3 y [0, 10] y [3, 10] y [7, 10] NC & AC Example courtesy of M. Fromherz In the temporal problem below • AC propagates bound • Unary constraint x>3 is imposed • AC propagates bounds again Consistency: Properties & Algorithms**Complexity of AC-1**Procedure AC-1: 1 begin 2 for i 1 untilndo NC(i) 3 Q {(i, j) | (i,j) arcs(G), i j 4 repeat 5 begin 6 CHANGE false 7 for each (i, j) Q do CHANGE (REVISE(i, j) or CHANGE) 8 end 9 until ¬ CHANGE 10 end Note: Q is not modified and |Q| = 2e • 4 9 repeats at most n·a times • Each iteration has |Q| = 2e calls to REVISE • Revise requires at most a2 checks of Pij AC-1 is O(a3· n · e) Consistency: Properties & Algorithms**AC versions**• AC-1 does not update Q, the queue of arcs • AC-2 iterates over arcs connected to at least one node whose domain has been modified. Nodes are ordered. • AC-3 same as AC-2, nodes are not ordered Consistency: Properties & Algorithms**Arc consistency (AC-3)**AC-3 iterates over arcs connected to at least one node whose domain has been modified Procedure AC-3: 1 begin 2 fori 1 untilndo NC(i) 3 Q { (i, j) | (i, j) arcs(G), i j } 4 WhileQ is not empty do 5 begin 6 select and delete any arc (k, m) from Q 7 If Revise(k, m) thenQQ { (i, k) | (i, k) arcs(G), i k, i m } 8 end 9 end Consistency: Properties & Algorithms**Procedure AC-3:**1 begin 2 fori 1 untilndo NC(i) 3 Q {(i, j) | (i, j) arcs(G), i j } 4 WhileQ is not empty do 5 begin select and delete any arc (k, m) from Q 7 If Revise(k, m) thenQQ { (i, k) | (i, k) arcs(G), i k, i m } 8 end 9 end First |Q| = 2e, then it grows and shrinks 48 Worst case: 1 element is deleted from DVk per iteration none of the arcs added is in Q Line 7: a·(dk - 1) Lines 4 - 8: Revise: a2 checks of Pij AC-3 is O(a2(2e + a(2e-n))) Connected graph (en – 1), AC-3 is O(a3e) If AC-p, AC-3 is O(a2e) AC-3 is (a2e) Complete graph: O(a3n2) Complexity of AC-3 Consistency: Properties & Algorithms**V2**V2 V1 V1 { 1, 2, 3, 4 } { 1, 2, 3, 4, 5 } { 1, 3, 5 } { 1, 2, 3, 4, 5 } { 1, 3, 5 } { 1, 2, 3, 5 } { 1, 2, 3, 4, 5 } { 1, 2, 3, 4, 5 } V4 V3 V4 V3 Example: Apply AC-3 Example: Apply AC-3 Thanks to Xu Lin DV1 = {1, 2, 3, 4, 5} DV2 = {1, 2, 3, 4, 5} DV3 = {1, 2, 3, 4, 5} DV4 = {1, 2, 3, 4, 5} CV2,V3 = {(2, 2), (4, 5), (2, 5), (3, 5), (2, 3), (5, 1), (1, 2), (5, 3), (2, 1), (1, 1)} CV1,V3 = {(5, 5), (2, 4), (3, 5), (3, 3), (5, 3), (4, 4), (5, 4), (3, 4), (1, 1), (3, 1)} CV2,V4 = {(1, 2), (3, 2), (3, 1), (4, 5), (2, 3), (4, 1), (1, 1), (4, 3), (2, 2), (1, 5)} Consistency: Properties & Algorithms**Applying AC-3 Thanks to Xu Lin**• Queue = {CV2, V4, CV4,V2, CV1,V3, CV2,V3, CV3, V1, CV3,V2} • Revise(V2,V4): DV2DV2 \ {5} = {1, 2, 3, 4} • Queue = {CV4,V2, CV1,V3, CV2,V3, CV3, V1, CV3, V2} • Revise(V4, V2): DV4 DV4 \ {4} = {1, 2, 3, 5} • Queue = {CV1,V3, CV2,V3, CV3, V1, CV3, V2} • Revise(V1, V3): DV1 {1, 2, 3, 4, 5} • Queue = {CV2,V3, CV3, V1, CV3, V2} • Revise(V2, V3): DV2 {1, 2, 3, 4} Consistency: Properties & Algorithms**Applying AC-3 (cont.) Thanks to Xu Lin**• Queue = {CV3, V1, CV3, V2} • Revise(V3, V1): DV3 DV3 \ {2} = {1, 3, 4, 5} • Queue = {CV2, V3, CV3, V2} • Revise(V2, V3): DV2 DV2 • Queue = {CV3, V2} • Revise(V3, V2): DV3 {1, 3, 5} • Queue = {CV1, V3} • Revise(V1, V3): DV1 {1, 3, 5} END Consistency: Properties & Algorithms**Main Improvements**Mohr & Henderson (AIJ 86): • AC-3 O(a3e) AC-4 O(a2e) • AC-4 is optimal • Trade repetition of consistency-check operations with heavy bookkeeping on which and how many values support a given value for a given variable • Data structures it uses: • m: values that are active, not filtered • s: lists all vvp's that support a given vvp • counter: given a vvp, provides the number of support provided by a given variable • How it proceeds: • Generates data structures • Prepares data structures • Iterates over constraints while updating support in data structures Consistency: Properties & Algorithms**counter**( V , V ), 3 0 4 2 ( V , V ), 2 0 4 2 ( V , V ), 1 0 4 2 ( V , V ), 1 0 2 3 M M ( V , V ), 4 0 1 3 Step 1: Generate 3 data structures • mand s have as many rows as there are vvp’s in the problem • counterhasas many rows as there are tuples in the constraints Consistency: Properties & Algorithms**Step 2: Prepare data structures**Data structures: s and counter. • Checks for every constraint CVi,Vj all tuples Vi=ai, Vj=bj_) • When the tuple is allowed, then update: • s(Vj,bj) s(Vj,bj) {(Vi, ai)} and • counter(Vj,bj) (Vj,bj) + 1 Update counter ((V2, V3), 2) to value 1 Update counter ((V3, V2), 2) to value 1 Update s-htable (V2, 2) to value ((V3, 2)) Update s-htable (V3, 2) to value ((V2, 2)) Update counter ((V2, V3), 4) to value 1 Update counter ((V3, V2), 5) to value 1 Update s-htable (V2, 4) to value ((V3, 5)) Update s-htable (V3, 5) to value ((V2, 4)) Consistency: Properties & Algorithms**Constraints CV2,V3 and CV3,V2**Updating m Note that (V3, V2),4 0 thus we remove 4 from the domain of V3 and update (V3, 4) nil in m Updating counter Since 4 is removed from DV3 then for every (Vk, l) | (V3, 4) s[(Vk, l)], we decrement counter[(Vk, V3), l] by 1 Consistency: Properties & Algorithms**Summary of arc-consistency algorithms**• AC-4 is optimal (worst-case) [Mohr & Henderson, AIJ 86] • Warning: worst-case complexity is pessimistic. Better worst-case complexity: AC-4 Better average behavior: AC-3 [Wallace, IJCAI 93] • AC-5: special constraints [Van Hentenryck, Deville, Teng 92] functional, anti-functional, and monotonic constraints • AC-6, AC-7: general but rely heavily on data structures for bookkeeping [Bessière & Régin] • Now, back to AC-3: AC-2000, AC-2001, AC-3.1, AC3.3 • Non-binary constraints: • GAC (general) [Mohr & Masini 1988], • all-different (dedicated) [Régin, 94] Consistency: Properties & Algorithms**Outline**• Motivation and material • Node and arc consistency and their complexity • Criteria for performance comparison & CSP parameters. • Which AC algorithm to choose? Project results may tell! • Path consistency and its complexity • K-consistency, minimality and decomposability • Other related results: box consistency, higher levels of consistency, (i, j)-consistency, etc. Consistency: Properties & Algorithms**CSP parameters ‹n, a, t, d›**• n is number of variables • a is maximum domain size • t is constraint tightness: • d is constraint density where e is the #constraints, emin=(n-1), and emax = n(n-1)/2 Lately, we use constraint ratiop = e/emax → Constraints in random problems often generated uniform → Use only connected graphs (throw the unconnected ones away) Consistency: Properties & Algorithms**Criteria for performance comparison**• Bounding time and space complexity (theoretical) • worst-case, • average-case, • best- case • Counting the number of constraint checks (theoretical, empirical) • Measuring CPU time (empirical) Consistency: Properties & Algorithms**Performance comparisonCourtesy of Lin XU**AC-4 AC-7 AC-3 Consistency: Properties & Algorithms**Wisdom (?) Free adaptation from Bessière**• When a single constraint check is very expensive to make, use AC-7. • When there is a lot of propagation, use AC-4 • When little propagation and constraint checks are cheap, use AC-3x Advice?Instructor's personal opinion Use AC-3.. Consistency: Properties & Algorithms**Outline**• Motivation and material • Node and arc consistency and their complexity • Criteria for performance comparison and CSP parameters. • Which AC algorithm to choose? Project results may tell! • Path consistency and its complexity • K-consistency, minimality and decomposability • Other related results: box consistency, higher levels of consistency, (i, j)-consistency, etc. Consistency: Properties & Algorithms**=**= = AC is not enough Example borrowed from Dechter Arc-consistent? Satisfiable? seek higher levels of consistency V V 1 1 b a a b V V V V 2 3 2 a b 3 b a a b a b Consistency: Properties & Algorithms**V2**V1 Vm-1 Vm V0 for all y DVm for all x DV0 Consistency of a path A path (V0, V1, V2, …, Vm) of length m is consistent iff • for any value xDV0 and for any value yDVm that are consistent (i.e., PV0 Vm(x, y)) • a sequence of values z1, z2, … , zm-1 in the domains of variables V1, V2, …, Vm-1, such that all constraints between them (along the path, not across it) are satisfied (i.e., PV0 V1(x, z1) PV1 V2(z1, z2) … PVm-1 Vm(zm-1, zm) ) Consistency: Properties & Algorithms**V2**V1 Vm-1 Vm V0 for all y DVm for all x DV0 Note The same variable can appear more than once in the path Every time, it may have a different value Constraints considered: PV0,Vm and those along the path All other constraints are neglected Consistency: Properties & Algorithms**V2**V3 V1 V4 {a, b, c} {a, b, c} {a, b, c} {a, b, c} All mutex constraints {a, b, c} {a, b, c} V5 V7 {a, b, c} V6 Example: consistency of a path Check path length = 2, 3, 4, 5, 6, .... Consistency: Properties & Algorithms**Path consistency: definition** A path of length m is path consistent A CSP is path consistent Property of a CSP Definition: A CSP is path consistent (PC) iff every path is consistent (i.e., any length of path) Question: should we enumerate every path of any length? Answer: No, only length 2, thanks to [Mackworth AIJ'77] Consistency: Properties & Algorithms**Making a CSP Path Consistent (PC)**Special case: Complete graph Theorem: In a complete graph, if every path of length 2 is consistent, the network is path consistent [Mackworth AIJ'77] PC-1: two operations, composition and intersection Proof by induction. Special case: Triangulated graph Theorem: In a triangulated graph, if every path of length 2 is consistent, the network is path consistent [Bliek & Sam-Haroud ‘99] PPC (partially path consistent) PC Consistency: Properties & Algorithms**Tools for PC-1**Two operators • Constraint composition: ( • ) R13 = R12• R23 • Constraint intersection: ( ) R13 R13, old R13, induced Consistency: Properties & Algorithms**Path consistency (PC-1)**Achieved by composition and intersection (of binary relations expressed as matrices) over all paths of length two. Procedure PC-1: 1 Begin 2 Yn R 3 repeat 4 begin 5 Y0 Yn 6 For k 1 untilndo 7 For i 1 untilndo 8 For j 1 untilndo 9 Ylij Yl-1ij Yl-1ik• Yl-1kk• Yl-1kj 10 end 11 until Yn = Y0 12 Y Yn 10 end Consistency: Properties & Algorithms