1. Soft constraint processing Thomas Schiex INRA – Toulouse France Javier Larrosa UPC – Barcelona Spain These slides are provided as a teaching support for the community. They can be freely modified and used as far as the original authors (T. Schiex and J. Larrosa) contribution is clearly mentionned and visible and that any modification is acknowledged by the author of the modification.

2. September 2006CP062 Overview 1.Frameworks Generic and specific 2.Algorithms Search: complete and incomplete Inference: complete and incomplete 3.Integration with CP Soft as hard Soft as global constraint

3. September 2006CP063 Parallel mini-tutorial  CSP  SAT strong relation  Along the presentation, we will highlight the connections with SAT  Multimedia trick: SAT slides in yellow background

4. September 2006CP064  CSP framework: natural for decision problems  SAT framework: natural for decision problems with boolean variables  Many problems are constrained optimization problems and the difficulty is in the optimization part Why soft constraints?

5. September 2006CP065 Why soft constraints? Given a set of requested pictures (of different importance)… … select the best subset of compatible pictures … … subject to available resources:  3 on-board cameras  Data-bus bandwith, setup-times, orbiting Best = maximize sum of importance  Earth Observation Satellite Scheduling

6. September 2006CP066 Why soft constraints?  Frequency assignment Given a telecommunication network …find the best frequency for each communication link avoiding interferences Best can be:  Minimize the maximum frequency (max)  Minimize the global interference (sum)

7. September 2006CP067 Why soft constraints?  Combinatorial auctions Given a set G of goods and a set B of bids…  Bid (b i,v i ), b i requested goods, v i value … find the best subset of compatible bids Best = maximize revenue (sum) G1G1 G3G3 G2G2 G5G5 G6G6 G4G4 G8G8 G7G7 b1v1b1v1 b4v4b4v4 b 2 v 2 b 3 v 3

8. September 2006CP068 Why soft constraints?  Probabilistic inference (bayesian nets) Given a probability distribution defined by a DAG of conditional probability tables and some evidence … …find the most probable explanation for the evidence (product)

9. September 2006CP069 Why soft constraints?  Even in decision problems: users may have preferences among solutions Experiment: give users a few solutions and they will find reasons to prefer some of them.

10. September 2006CP0610 Observation  Optimization problems are harder than satisfaction problems CSP vs. Max-CSP

11. September 2006CP0611 Why is it so hard ? Problem P(alpha): is there an assignment of cost lower than alpha ? Proof of inconsistency Proof of optimality Harder than finding an optimum

12. September 2006CP0612 Notation  X={x 1,..., x n } variables ( n variables)  D={D 1,..., D n } finite domains (max size d )  Z⊆Y⊆X, t Y is a tuple on Y t Y [Z] is its projection on Z t Y [-x] = t Y [Y-{x}]is projecting out variable x f Y : ∏ x i ∊Y D i  Eis a cost function on Y

13. Generic and specific frameworks Valued CNweighted CN Semiring CNfuzzy CN …

14. September 2006CP0614 Costs (preferences)  E costs (preferences) set ordered by ≼ if a ≼ b then a is better than b  Costs are associated to tuples  Combined with a dedicated operator max: priorities +: additive costs *: factorized probabilities… Fuzzy/possibilistic CN Weighted CN Probabilistic CN, BN 

15. September 2006CP0615 Soft constraint network (CN) (X,D,C)(X,D,C) X={x 1,..., x n } variables D={D 1,..., D n } finite domains C={f,...} cost functions  f S, f ij, f i f ∅ scope S,{x i,x j },{x i }, ∅  f S (t):  E (ordered by ≼,  ≼ T )  Obj. Function: F(X)=  f S (X[S])  Solution: F(t)  T  Task: find optimal solution identity commutative associative monotonic anihilator

16. September 2006CP0616 Specific frameworks Instance E  ≼T≼T Classic CN {t,f}{t,f} and t ≼ f Possibilistic [0,1] max 0≼10≼1 Fuzzy CN [0,1] max ≼ 1≼01≼0 Weighted CN [0,k]+ 0≼k0≼k Bayes net [0,1] × 1≼01≼0

17. September 2006CP0617 Weighted Clauses  (C,w)weighted clause Cdisjunction of literals wcost of violation w  E (ordered by ≼,  ≼ T)  combinator of costs  Cost functions = weighted clauses xixi xjxj f(x i,x j ) 006 010 102 113 (x i ν x j, 6), ( ¬ x i ν x j, 2), ( ¬ x i ν ¬ x j, 3)

18. September 2006CP0618 Soft CNF formula  F={(C,w),…} Set of weighted clauses  (C, T )mandatory clause  (C, w< T )non-mandatory clause  Valuation: F(X)=  w (aggr. of unsatisfied)  Model: F(t)  T  Task: find optimal model

19. September 2006CP0619 Specific weighted prop. logics Instance E  ≼T≼T SAT {t,f}{t,f} and t ≼ f Fuzzy SAT [0,1] max ≼ 1≼01≼0 Max-SAT [0,k]+ 0≼k0≼k Markov Prop. Logic [0,1] × 1≼01≼0

20. September 2006CP0620 CSP example (3-coloring) x3x3 x2x2 x5x5 x1x1 x4x4 For each edge: (hard constr.) xixi xjxj f(x i,x j ) bbT bg  br  gb  ggT gr  rb  rg  rrT

21. September 2006CP0621 Weighted CSP example (  = +) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): number of non blue vertices For each vertex xixi f(x i ) b0 g1 r1

22. September 2006CP0622 Possibilistic CSP example (  =max) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): highest color used (b<g<r) For each vertex xixi f(x i ) b0.0 g0.1 r0.2

23. September 2006CP0623 Some important details  T = maximum acceptable violation.  Empty scope soft constraint f  (a constant) Gives an obvious lower bound on the optimum If you do not like it: f  =  Additional expression power

24. September 2006CP0624 Weighted CSP example (  = +) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): number of non blue vertices For each vertex For each edge: xixi f(x i ) b0 g1 r1 xixi xjxj f(x i,x j ) bbT bg0 br0 gb0 ggT gr0 rb0 rg0 rrT T=6 f  = 0 T=3 Optimal coloration with less than 3 non-blue

25. September 2006CP0625 General frameworks and cost structures hard {,T} totally ordered Semiring CSP Valued CSP idempotent fair multi criteria lattice ordered multiple

26. September 2006CP0626 Idempotency a  a = a (for any a ) For any f S implied by (X,D,C) (X,D,C) ≡ (X,D,C∪{f S }) Classic CN:  = and Possibilistic CN:  = max Fuzzy CN:  = max ≼ …

27. September 2006CP0627 Fairness  Ability to compensate for cost increases by subtraction using a pseudo-difference: For b ≼ a, (a ⊖ b)  b = a Classic CN: a ⊖b = or (max) Fuzzy CN: a⊖b = max ≼ Weighted CN: a⊖b = a-b (a≠T) else T Bayes nets: a⊖b = / …

28. Processing Soft constraints Search complete (systematic) incomplete (local) Inference complete (variable elimination) incomplete (local consistency)

29. Systematic search Branch and bound(s)

30. September 2006CP0630 I - Assignment (conditioning) xixi xjxj f(x i,x j ) bbT bg0 br3 gb0 ggT gr0 rb0 rg0 rrT f[x i =b] xjxj bT g0 r3 g(x j ) g[x j =r] 0 3 hh

31. September 2006CP0631 I - Assignment (conditioning) {(x  y  z,3), (¬x  y,2)} x=true (y,2)(,2) y=false empty clause. It cannot be satisfied, 2 is necessary cost

32. September 2006CP0632 Systematic search (LB) Lower Bound (UB) Upper Bound If  then prune variables under estimation of the best solution in the sub-tree = best solution so far Each node is a soft constraint subproblem LBff = f  = T UBT

33. September 2006CP0633 Depth First Search (DFS) BT(X,D,C) if ( X=  ) then Top :=f  else x j := selectVar(X) forall a  D j do  f S  C s.t. x j  S f := f[x j =a] f  :=  g S  C s.t. S=  g S if ( f  <Top) then BT(X-{x j },D-{D j },C ) variable heuristics improve LB good UB ASAP value heuristics

34. September 2006CP0634 Improving the lower bound (WCSP)  Sum up costs that will necessarily occur (no matter what values are assigned to the variables)  PFC-DAC (Wallace et al. 1994)  PFC-MRDAC (Larrosa et al. 1999…)  Russian Doll Search (Verfaillie et al. 1996)  Mini-buckets (Dechter et al. 1998)

35. September 2006CP0635 Improving the lower bound (Max-SAT)  Detect independent subsets of mutually inconsistent clauses  LB4a (Shen and Zhang, 2004)  UP (Li et al, 2005)  Max Solver (Xing and Zhang, 2005)  MaxSatz (Li et al, 2006)  …

36. Local search Nothing really specific

37. September 2006CP0637 Local search Based on perturbation of solutions in a local neighborhood  Simulated annealing  Tabu search  Variable neighborhood search  Greedy rand. adapt. search (GRASP)  Evolutionary computation (GA)  Ant colony optimization…  See: Blum & Roli, ACM comp. surveys, 35(3), 2003 For boolean variables: GSAT …

38. September 2006CP0638 Boosting Systematic Search with Local Search Local search (X,D,C) time limit Sub-optimal solution  Do local search prior systematic search  Use best cost found as initial T If optimal, we just prove optimality In all cases, we may improve pruning

39. September 2006CP0639 Boosting Systematic Search with Local Search  Ex: Frequency assignment problem Instance: CELAR6-sub4  #var: 22, #val: 44, Optimum: 3230 Solver: toolbar 2.2 with default options T initialized to 100000  3 hours T initialized to 3230  1 hour  Optimized local search can find the optimum in a less than 30” ( incop )

40. Complete inference Variable (bucket) elimination Graph structural parameters

41. September 2006CP0641 II - Combination (join with , + here ) xixi xjxj f(x i,x j ) bb6 bg0 gb0 gg6 xjxj xkxk g(x j,x k ) bb6 bg0 gb0 gg6 xixi xjxj xkxk h(x i,x j,x k ) bbb12 bbg6 bgb0 bgg6 gbb6 gbg0 ggb6 ggg  = 0  6

42. September 2006CP0642 III - Projection (elimination) xixi xjxj f(x i,x j ) bb4 bg6 br0 gb2 gg6 gr3 rb1 rg0 rr6 f[x i ] xixi g(x i ) b g r 0 0 2 g[  ] hh 0 Min

43. September 2006CP0643 Properties  Replacing two functions by their combination preserves the problem  If f is the only function involving variable x, replacing f by f[-x] preserves the optimum

44. September 2006CP0644 Variable elimination 1.Select a variable 2.Sum all functions that mention it 3.Project the variable out Complexity Time:  (exp(deg+1)) Space:  (exp(deg))

45. September 2006CP0645 Variable elimination (aka bucket elimination)  Eliminate Variables one by one.  When all variables have been eliminated, the problem is solved  Optimal solutions of the original problem can be recomputed Complexity: exponential in the induced width

46. September 2006CP0646 Elimination order influence  {f(x,r), f(x,z), …, f(x,y)}  Order: r, z, …, y, x x rzy …

47. September 2006CP0647 Elimination order influence  {f(x,r), f(x,z), …, f(x,y)}  Order: r, z, …, y, x x rzy …

48. September 2006CP0648 Elimination order influence  {f(x), f(x,z), …, f(x,y)}  Order: z, …, y, x x zy …

49. September 2006CP0649 Elimination order influence  {f(x), f(x,z), …, f(x,y)}  Order: z, …, y, x x zy …

50. September 2006CP0650 Elimination order influence  {f(x), f(x), f(x,y)}  Order: y, x x y …

51. September 2006CP0651 Elimination order influence  {f(x), f(x), f(x,y)}  Order: y, x x y …

52. September 2006CP0652 Elimination order influence  {f(x), f(x), f(x)}  Order: x x

53. September 2006CP0653 Elimination order influence  {f(x), f(x), f(x)}  Order: x x

54. September 2006CP0654 Elimination order influence  {f()}  Order:

55. September 2006CP0655 Elimination order influence  {f(x,r), f(x,z), …, f(x,y)}  Order: x, y, z, …, r x rzy …

56. September 2006CP0656 Elimination order influence  {f(x,r), f(x,z), …, f(x,y)}  Order: x, y, z, …, r x rzy …

57. September 2006CP0657 Elimination order influence  {f(r,z,…,y)}  Order: y, z, r rzy … CLIQUE

58. September 2006CP0658 Induced width  For G=(V,E) and a given elimination (vertex) ordering, the largest degree encountered is the induced width of the ordered graph  Minimizing induced width is NP-hard.

59. September 2006CP0659 History / terminology  SAT: Directed Resolution (Davis and Putnam, 60)  Operations Research: Non serial dynamic programming (Bertelé Brioschi, 72)  Databases: Acyclic DB (Beeri et al 1983)  Bayesian nets: Join-tree (Pearl 88, Lauritzen et Spiegelhalter 88)  Constraint nets: Adaptive Consistency (Dechter and Pearl 88)

60. September 2006CP0660 Boosting search with variable elimination: BB-VE(k)  At each node Select an unassigned variable x i If deg i ≤ k then eliminate x i Else branch on the values of x i  Properties BE-VE(-1) is BB BE-VE(w*) is VE BE-VE(1) is similar to cycle-cutset

61. September 2006CP0661 Boosting search with variable elimination  Ex: still-life (academic problem) Instance: n=14  #var:196, #val:2 Ilog Solver  5 days Variable Elimination  1 day BB-VE(18)  2 seconds

62. September 2006CP0662 Memoization fights thrashing = Different nodes, Same subproblem store retrieve Detecting subproblems equivalence is hard VV t P t’t’ P’P’ P t t’t’

63. September 2006CP0663 Context-based memoization  P=P’, if |t|=|t’| and same assign. to partially assigned cost functions t t’t’ PP’P’

64. September 2006CP0664 Memoization  Depth-first B&B with, context-based memoization independent sub-problem detection  … is essentialy equivalent to VE Therefore space expensive  Fresh approach: Easier to incorporate typical tricks such as propagation, symmetry breaking,…  Algorithms: Recursive Cond. (Darwiche 2001) BTD (Jégou and Terrioux 2003) AND/OR (Dechter et al, 2004) Adaptive memoization: time/space tradeoff

65. September 2006CP0665 SAT inference  In SAT, inference = resolution x  A ¬xB¬xB ------------ A  B  Effect: transforms explicit knowledge into implicit  Complete inference: Resolve until quiescence Smart policy: variable by variable (Davis & Putnam, 60). Exponential on the induced width.

66. September 2006CP0666 Fair SAT Inference (x  A,u), (¬x  B,w)  (A  B,m), (x  A,u ⊖ m), (¬x  B, w ⊖ m), (x  A  ¬B,m), (¬x  ¬A  B,m) where: m=min{u,w} Effect: moves knowledge

67. September 2006CP0667 Example: Max-SAT (  =+, ⊖ =-) (x  y,3), (  x  z,3) = xx y yy z x xx y zz yy z x 3 3 zz 3 3 3 (y  z,3), (x  y,3-3), (  x  z,3-3), (x  y  z,3), (  x  y  z,3)

68. September 2006CP0668 Properties (Max-SAT)  In SAT, collapses to classical resolution  Sound and complete  Variable elimination: Select a variable x Resolve on x until quiescence Remove all clauses mentioning x  Time and space complexity: exponential on the induced width

69. September 2006CP0669 Change

70. Incomplete inference Local consistency Restricted resolution

71. September 2006CP0671 Incomplete inference  Tries to trade completeness for space/time Produces only specific classes of cost functions Usually in polynomial time/space  Local consistency: node, arc… Equivalent problem Compositional: transparent use Provides a lb on consistencyoptimal cost

72. September 2006CP0672 Classical arc consistency  A CSP is AC iff for any x i and c ij c i = c i ⋈ (c ij ⋈ c j )[x i ] namely, (c ij ⋈ c j )[x i ] brings no new information on x i w v v w 0 0 0 0 ij T T xixi xjxj c ij vv0 vw0 wvT wwT xixi c(x i ) v0 wT c ij ⋈ c j (c ij ⋈ c j )[x i ]

73. September 2006CP0673 Enforcing AC  for any x i and c ij c i := c i ⋈ (c ij ⋈ c j )[x i ] until fixpoint (unique) w v v w 0 0 0 0 ij T T xixi xjxj c ij vv0 vw0 wvT wwT xixi c(x i ) v0 wT c ij ⋈ c j T (c ij ⋈ c j )[x i ]

74. September 2006CP0674 Arc consistency and soft constraints w v v w 0 0 0 0 ij 2 1 xixi xjxj f ij vv0 vw0 wv2 ww1 XiXi f(x i ) v0 w1 f ij  f j 1 Always equivalent iff  idempotent  for any x i and f ij f=(f ij  f j )[x i ] brings no new information on x i (f ij  f j )[x i ]

75. September 2006CP0675 Idempotent soft CN  The previous operational extension works on any idempotent semiring CN Chaotic iteration of local enforcing rules until fixpoint Terminates and yields an equivalent problem Extends to generalized k-consistency Total order: idempotent  (  = max)

76. September 2006CP0676 Non idempotent: weighted CN  for any x i and f ij f=(f ij  f j )[x i ] brings no new information on x i w v v w 0 0 0 0 ij 2 1 xixi xjxj f ij vv0 vw0 wv2 ww1 xixi f(x i ) v0 w1 f ij  f j 1 EQUIVALENCE LOST f ij  f j [x i ]

77. September 2006CP0677  Combination+Subtraction: equivalence preserving transformation IV - Subtraction of cost functions (fair) 2 w v v w 0 0 0 ij 1 01 1

78. September 2006CP0678 (K,Y) equivalence preserving inference  For a set K of cost functions and a scope Y Replace K by (  K) Add (  K)[Y] to the CN (implied by  K) Subtract (  K)[Y] from (  K)  Yields an equivalent network  All implicit information on Y in K is explicit  Repeat for a class of (K,Y) until fixpoint

79. September 2006CP0679 Node Consistency (NC * ): ({f ,f i },  ) EPI For any variable X i   a, f  + f i (a)<T   a, f i (a)= 0 Complexity: O(nd) w v v v w w f =f = T = 3 2 2 1 1 1 1 1 0 0 1 x y z 0 0 0 1 4 Or T may decrease: back-propagation

80. September 2006CP0680 0 1 Full AC (FAC * ): ({f ij,f j },{x i }) EPI NC * For all f ij   a  b f ij (a,b) + f j (b) = 0 (full support) w v v w f  =0 T=4 0 1 0 1 0 x z 1 1 That’s our starting point! No termination !!!

81. September 2006CP0681 0 Arc Consistency (AC * ): ({f ij },{x i }) EPI NC * For all f ij   a  b f ij (a,b)= 0 b is a support complexity: O(n 2 d 3 ) w v v w w f =f = T=4 2 1 1 1 0 0 0 0 1 x y z 1 1 2 0

82. September 2006CP0682 Neighborhood Resolution (x  A,u), (¬x  A,w)  (A,m), (x  A,u ⊖ m), (¬x  A, w ⊖ m), (x  A  ¬A,m), (¬x  ¬A  A,m) if |A|=0, enforces node consistency if |A|=1, enforces arc consistency

83. September 2006CP0683 Confluence is lost w 0 1 yx v w 0 0 v 1 f  = 0 1

84. September 2006CP0684 Confluence is lost Finding an AC closure that maximizes the lb is an NP-hard problem (Cooper & Schiex 2004). Well… one can do better in pol. time (OSAC, IJCAI 2007) w 0 1 yx v w 0 0 v f  = 0 1

85. September 2006CP0685 Hierarchy NC * O(nd) AC * O(n 2 d 3 )DAC * O(ed 2 ) FDAC * O(end 3 ) AC NC DAC Special case: CSP (Top=1) EDAC * O(ed 2 max{nd,T})

86. September 2006CP0686 Boosting search with LC BT(X,D,C) if (X=  ) then Top :=f  else x j := selectVar(X) forall a  D j do  f S  C s.t. x j  S f S := f S [x j =a] if (LC) then BT(X-{x j },D-{D j },C)

87. September 2006CP0687 BT MNC MAC/MDAC MFDAC MEDAC

88. September 2006CP0688 Boosting Systematic Search with Local consistency Frequency assignment problem  CELAR6-sub4 (22 var, 44 val, 477 cost func): MNC*  1 year MFDAC*  1 hour  CELAR6 (100 var, 44 val, 1322 cost func): MEDAC+memoization  3 hours (toolbar-BTD)

89. September 2006CP0689 Beyond Arc Consistency  Path inverse consistency PIC (Debryune & Bessière) xy z (x,a) can be pruned because there are two other variables y,z such that (x,a) cannot be extended to any of their values. ({f y, f z, f xy, f xz, f yz },{x}) EPI a b c a b c a b c

90. September 2006CP0690 Beyond Arc Consistency  Soft Path inverse consistency PIC* 2 2 3 xyz aaa2 aab5 aba2 abb3 baa0 bab0 bba2 bbb0 ({f y, f z, f xy, f xz, f yz },x) EPI x a2 b0 f y  f z  f xy  f xz  f yz (f y f z f xy f xz f yz )[x] x y z a b a b a b xyz aaa0 aab3 aba0 abb1 baa0 bab0 bba2 bbb0 1 2 2

91. September 2006CP0691 Hyper-resolution (2 steps) (l  h  A,u), (  l  q  A,v), (  h  q  A,u) (h  q  A,m), (l  h  A,u-m), (  l  q  A,v-m), (l  h  q  A,m), (  l  q  h  A,m), (  h  q  A,u) (q  A,m’), (h  q  A,m-m’), (  h  q  A,u-m’), (l  h  A,u-m), (  l  q  A,v-m), (l  h  q  A,m), (  l  q  h  A,m) if |A|=0, equal to soft PIC Impressive empirical speed-ups ==

92. Complexity & Polynomial classes Tree = induced width 1 Idempotent  or not…

93. September 2006CP0693 Polynomial classes Idempotent VCSP: min-max CN  Can use  -cuts for lifting CSP classes Sufficient condition: the polynomial class is «conserved» by  -cuts Simple TCSP are TCSP where all constraints use 1 interval: x i -x j ∊[a ij,b ij ] Fuzzy STCN: any slice of a cost function is an interval (semi-convex function) (Rossi et al.)

94. September 2006CP0694 Hardness in the additive case (weighted/boolean)  MaxSat is MAXSNP complete (no PTAS) Weighted MaxSAT is FP NP - complete MaxSAT is FP NP [O(log(n))] complete: weights ! MaxSAT tractable langages fully characterized (Creignou 2001)  MaxCSP langage: f eq (x,y) : (x = y) ? 0 : 1 is NP-hard. Submodular cost function lang. is polynomial. (u ≤ x, v ≤ y f(u,v)+f(x,y) ≤ f(u,y)+f(x,v)) (Cohen et al.)

95. Integration of soft constraints into classical constraint programming Soft as hard Soft local consistency as a global constraint

96. September 2006CP0696 Soft constraints as hard constraints  one extra variable x s per cost function f S  all with domain E  f S ➙ c S ∪{x S } allowing (t,f S (t)) for all t∊ℓ(S)  one variable x C =  x s (global constraint)

97. September 2006CP0697 Soft as Hard (SaH)  Criterion represented as a variable  Multiple criteria = multiple variables  Constraints on/between criteria  Weaknesses: Extra variables (domains), increased arities SaH constraints give weak GAC propagation Problem structure changed/hidden

98. September 2006CP0698 ≥ Soft AC « stronger than » SasH GAC  Take a WCSP  Enforce Soft AC on it  Each cost function contains at least one tuple with a 0 cost (definition)  Soft as Hard: the cost variable x C will have a lb of 0  The lower bound cannot improve by GAC

99. September 2006CP0699 1 1 1 1 1 1 f ∅ =1 a b x1x1 x3x3 x2x2 x1x1 x3x3 x2x2 1 0 1 0 1 0 x 12 x 23 xcxc x c =x 12 +x 23 2 > Soft AC « stronger than » SasH GAC

100. September 2006CP06100 Soft local Consistency as a Global constraint (  =+)  Global constraint: Soft(X,F,C) Xvariables Fcost functions Cinterval cost variable (ub = T)  Semantics: X U{C} satisfy Soft(X,F,C) iff  f(X)=C  Enforcing GAC on Soft is NP-hard  Soft consistency: filtering algorithm (lb≥f  )

101. September 2006CP06101 Ex: Spot 5 (Earth satellite sched.)  For each requested photography: € lost if not taken, Mb of memory if taken  variables: requested photographies  domains: {0,1,2,3}  constraints: {r ij, r ijk } binary and ternary hard costraints Sum(X)<Cap. global memory bound Soft(X,F 1,€)bound € loss

102. September 2006CP06102 Example: soft quasi-group (motivated by sports scheduling)  Alldiff(x i1,…,x in ) i=1..m  Alldiff(x 1j,…,x mj ) j=1..n  Soft(X,{f ij },[0..k],+) 3 4 Minimize #neighbors of different parity Cost 1

103. Global soft constraints

104. September 2006CP06104 Global soft constraints  Idea: define a library of useful but non- standard objective functions along with efficient filtering algorithms AllDiff (2 semantics: Petit et al 2001, van Hoeve 2004 ) Soft global cardinality (van Hoeve et al. 2004) Soft regular (van Hoeve et al. 2004) … all enforce reified GAC

105. September 2006CP06105 Conclusion  A large subset of classic CN body of knowledge has been extended to soft CN, efficient solving tools exist.  Much remains to be done: Extension: to other problems than optimization (counting, quantification…) Techniques: symmetries, learning, knowledge compilation… Algorithmic: still better lb, other local consistencies or dominance. Global (SoftAsSoft). Exploiting problem structure. Implementation: better integration with classic CN solver (Choco, Solver, Minion…) Applications: problem modelling, solving, heuristic guidance, partial solving.

106. 30’’ of publicity

107. September 2006CP06107 Open source libraries Toolbar and Toulbar2  Accessible from the Soft wiki site : carlit.toulouse.inra.fr/cgi-bin/awki.cgi/SoftCSP  Alg: BE-VE,MNC,MAC,MDAC,MFDAC,MEDAC,MPIC,BTD  ILOG connection, large domains/problems…  Read MaxCSP/SAT (weighted or not) and ERGO format  Thousands of benchmarks in standardized format  Pointers to other solvers (MaxSAT/CSP)  Forge mulcyber.toulouse.inra.fr/projects/toolbar (toulbar2) mulcyber.toulouse.inra.fr/projects/toolbartoulbar2 Pwd: bia31

108. September 2006CP06108 Thank you for your attention This is it !  S. Bistarelli, U. Montanari and F. Rossi, Semiring-based Constraint Satisfaction and Optimization, Journal of ACM, vol.44, n.2, pp. 201-236, March 1997.Semiring-based Constraint Satisfaction and Optimization  S. Bistarelli, H. Fargier, U. Montanari, F. Rossi, T. Schiex, G. Verfaillie. Semiring- Based CSPs and Valued CSPs: Frameworks, Properties, and Comparison. CONSTRAINTS, Vol.4, N.3, September 1999.Semiring- Based CSPs and Valued CSPs: Frameworks, Properties, and Comparison  S. Bistarelli, R. Gennari, F. Rossi. Constraint Propagation for Soft Constraint Satisfaction Problems: Generalization and Termination Conditions, in Proc. CP 2000Constraint Propagation for Soft Constraint Satisfaction Problems: Generalization and Termination Conditions  C. Blum and A. Roli. Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Computing Surveys, 35(3):268-308, 2003.Metaheuristics in combinatorial optimization: Overview and conceptual comparison  T. Schiex, Arc consistency for soft constraints, in Proc. CP’2000.  M. Cooper, T. Schiex. Arc consistency for soft constraints, Artificial Intelligence, Volume 154 (1-2), 199-227 2004.Arc consistency for soft constraints  M. Cooper. Reduction Operations in fuzzy or valued constraint satisfaction problems. Fuzzy Sets and Systems 134 (3) 2003.  A. Darwiche. Recursive Conditioning. Artificial Intelligence. Vol 125, No 1-2, pages 5-41.  R. Dechter. Bucket Elimination: A unifying framework for Reasoning. Artificial Intelligence, October, 1999.  R. Dechter, Mini-Buckets: A General Scheme For Generating Approximations In Automated Reasoning In Proc. Of IJCAI97

109. September 2006CP06109 References  S. de Givry, F. Heras, J. Larrosa & M. Zytnicki. Existential arc consistency: getting closer to full arc consistency in weighted CSPs. In IJCAI 2005.Existential arc consistency: getting closer to full arc consistency in weighted CSPs  W.-J. van Hoeve, G. Pesant and L.-M. Rousseau. On Global Warming: Flow-Based Soft Global Constraints. Journal of Heuristics 12(4-5), pp. 347-373, 2006.  P. Jegou & C. Terrioux. Hybrid backtracking bounded by tree-decomposition of constraint networks. Artif. Intell. 146(1): 43-75 (2003)Artif. Intell. 146  J. Larrosa & T. Schiex. Solving Weighted CSP by Maintaining Arc Consistency. Artificial Intelligence. 159 (1-2): 1-26, 2004.Solving Weighted CSP by Maintaining Arc Consistency  J. Larrosa and T. Schiex. In the quest of the best form of local consistency for Weighted CSP, Proc. of IJCAI'03In the quest of the best form of local consistency for Weighted CSP  J. Larrosa, P. Meseguer, T. Schiex Maintaining Reversible DAC for MAX-CSP. Artificial Intelligence.107(1), pp. 149-163.  R. Marinescu and R. Dechter. AND/OR Branch-and-Bound for Graphical Models. In proceedings of IJCAI'2005.  J.C. Regin, T. Petit, C. Bessiere and J.F. Puget. An original constraint based approach for solving over constrained problems. In Proc. CP'2000.  T. Schiex, H. Fargier et G. Verfaillie. Valued Constraint Satisfaction Problems: hard and easy problems In Proc. of IJCAI 95.Valued Constraint Satisfaction Problems: hard and easy problems  G. Verfaillie, M. Lemaitre et T. Schiex. Russian Doll Search Proc. of AAAI'96.Russian Doll Search

110. September 2006CP06110 References  M. Bonet, J. Levy and F. Manya. A complete calculus for max-sat. In SAT 2006.  M. Davis & H. Putnam. A computation procedure for quantification theory. In JACM 3 (7) 1960.  I. Rish and R. Dechter. Resolution versus Search: Two Strategies for SAT. In Journal of Automated Reasoning, 24 (1-2), 2000.  F. Heras & J. Larrosa. New Inference Rules for Efficient Max-SAT Solving. In AAAI 2006.New Inference Rules for Efficient Max-SAT Solving  J. Larrosa, F. Heras. Resolution in Max-SAT and its relation to local consistency in weighted CSPs. In IJCAI 2005.Resolution in Max-SAT and its relation to local consistency in weighted CSPs  C.M. Li, F. Manya and J. Planes. Improved branch and bound algorithms for max- sat. In AAAI 2006.  H. Shen and H. Zhang. Study of lower bounds for max-2-sat. In proc. of AAAI 2004.  Z. Xing and W. Zhang. MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability. Artificial Intelligence 164 (1-2) 2005.

111. September 2006CP06111 SoftasHard GAC vs. EDAC 25 variables, 2 values binary MaxCSP  Toolbar MEDAC opt=34 220 nodes cpu-time = 0’’  GAC on SoftasHard, ILOG Solver 6.0, solve opt = 34 339136 choice points cpu-time: 29.1’’ Uses table constraints

112. September 2006CP06112 Other hints on SoftasHard GAC  MaxSAT as Pseudo Boolean  SoftAsHard For each clause: c = (x  …  z,p c ) c SAH = (x  …  z  r c ) Extra cardinality constraint: Σ p c.r c ≤ k Used by SAT4JMaxSat (MaxSAT competition).

113. September 2006CP06113 MaxSAT competition (SAT 2006) Unweighted MaxSAT

114. September 2006CP06114 MaxSAT competition (SAT 2006) Weighted