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Soft constraints

Soft constraints. Classical (hard) Constraints. Variables {X 1 ,…,X n }=X Domains {D(X 1 ),…,D(X n )}=D Constraints : each constraint c connects a subset of X and is a subset of the Cartesian product of the domains of its variables ( allowed combinations) Example: graph coloring

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Soft constraints

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  1. Soft constraints

  2. Classical (hard) Constraints • Variables{X1,…,Xn}=X • Domains{D(X1),…,D(Xn)}=D • Constraints: each constraint c connects a subset of X and is a subset of the Cartesian product of the domains of its variables (allowed combinations) • Example: graph coloring • Variables: nodes of the graph • Domains: sets of colours • Constraints: two variables, set of pairs of colours (different colours)     

  3. Classical constraints are useful for ... • resource allocation • scheduling • vehicle routing • car sequencing • timetabling • graph coloring • VLSI design • vision • Robotics • ... However, in most real-life situations we need to express fuzziness, possibilities, preferences, probabilities, costs, ...

  4. Why is this not enough? • Over-constrained problems • it does not make sense to just say that there is no solution • Optimization problems • Also multi-criteria • Problems with both preferences and hard statements • ex.: time-tabling (hard constraints about space and time, soft constraints about teachers) • Problems with uncertainty

  5. A problem with preferences • The problem: to decide what to eat at a restaurant, given some choices for drinks and dishes • The client has some preferences over drinks, dishes, and their combinations • Examples: • White wine is the most preferred drink • Pizza is the most preferred dish • Fish is less preferred • white wine and fish is the most preferred combination • red wine and meat is less preferred • We look for a combination which “maximizes” the overall preference of the client

  6. Soft Constraints: the C-semiring framework • Variables {X1,…,Xn}=X • Domains {D(X1),…,D(Xn)}=D • C-semiring <A,+,x,0,1>: • A set of preferences • + additive operator, inducing the ordering: a≥biff a+b=a (idempotent, commutative, associative, unit element 0); • x multiplicative operator: combines preferences (commutative, associative, unit element 1, absorbing element 0) • 0,1 respect. bottom and top element • x distributes over + [Bistarelli, Montanari, Rossi, IJCAI 1995]

  7. Soft Constraints: the C-semiring framework • + and x monotone on ≤ • for all a,b in A, a x b ≤ a • for all a in A, 0 ≤ a ≤ 1 • Lattice <A,≤> • + is lub • x glb if x idempotent • When ≤ is a total order and x idempotent: • if a ≤ b, then a + b = b and a x b = a, thus +=max and x=min

  8. Soft constraints • Soft constraint: a pair c=<f,con> where: • Scope: con={Xc1,…, Xck} subset of X • Preference function : f: D(Xc1)x…xD(Xck) → A tuple (v1,…, vk) → p preference • Hard constraint: a soft constraint where for each tuple (v1,…, vk) f (v1,…, vk)=1 the tuple is allowed f (v1,…, vk)=0 the tuple is forbidden

  9. Instances of soft constraints • Each instance is characterized by a c-semiring <A, +, x, 0, 1> • Classical constraints: <{0,1},or,and,0,1> • Fuzzy constraints: <[0,1],max,min,0,1> • Lexicographic CSPs: <[0,1]k,lex-max,min,0k,1k> • Weighted constraints (N):<N+,min,+,+,0> • Weighted constraints (R):<R+,min,+,+,0> • Max CSP: weight =1 when constraint is not satisfied and 0 is satisfied • Probabilistic constraints: <[0,1],max,x,0,1> • Valued CSPs: any semiring with a total order • Multicriteria problems: Cartesian product of semirings

  10. {s=slow, f=fast} {256, 512, 1024} VGA MB <s,256> → 1 <s,512> → 1 <s,1024> → 1 <f,256> → 0 <f,512> → 1 <f,1024> → 1 Example of hard constraint The fast VGA is compatible only with at least 512MB of memory

  11. {s=slow, f=fast} {256, 512, 1024} VGA MB <s,256> → 0.6 <s,512> → 0.7 <s,1024> → 0.9 <f,256> → 0.1 <f,512> → 0.9 <f,1024> → 1 Example of soft (fuzzy) constraint The VGA performs better with more memory • Soft constraints are a quantitative formalism

  12. Solution ordering • A soft CSP induces an ordering over the solutions, from the ordering of the semiring • Totally ordered semiring  total order over solutions (possibly with ties) • Partially ordered semiring  total or partial order over solutions (possibly with ties) • Any ordering can be obtained

  13. Expressive power • A B iff from a problem P in A it is possible to build in polynomial time a problem P’ in B s.t. the optimal solutions are the same(but not necessarily the solution ordering!) • A B iff from a problem P in A it is possible to build in polynomial time a problem P’ in B s.t. opt(P’)  opt(P)

  14. Semiring-based Prob Valued weightedR weightedN Fuzzy Lexicographic Classical Expressive power

  15. {s=slow, f=fast} {256, 512, 1024} VGA MB <s,256> → 0.6 <s,512> → 0.7 <s,1024> → 0.9 <f,256> → 0.1 <f,512> → 0.9 <f,1024> → 1 • Projection over VGA: • s max(0.6,0.7,0.9) = 0.9 • f  max(0.1,0.9,1) = 1 Fuzzy constraints: <[0,1],max,min,0,1> Projection If c=<f,con>, then c|I = <f', I  con> • f'(t') = +(f(t)) over tuples of values t s.t. t|I  con = t’

  16. 256  0.5 512  0.8 1024  0.7 s=slow 0.2 f=fast  0.9 P4  0.7 AMD  0.4 VGA MB P <s,256> → 0.6 <s,512> → 0.7 <s,1024> → 0.9 <f,256> → 0.1 <256,P4> → 0.5 <256,AMD> → 0.5 <f,512> → 0.9 <512,P4> → 0.7 <512,AMD> → 0.5 <f,1024> → 1 <1024,AMD> → 0.5 <1024,P4> → 0.9 f(s,256,P4) = min(0.2,0.6,0.5,0.5) = 0.2 f(f,1024,P4)=min(0.9,1,0.7,0.9,0.7)=0.7 Combination If ci=<fi,coni>, then c1 x c2 = <f, con1  con2> • f(t) = f1(t|con1) x f2(t|con2)

  17. 256  0.5 512  0.8 1024  0.7 s=slow 0.2 f=fast  0.9 P4  0.7 AMD  0.4 VGA MB P <s,256> → 0.6 <s,512> → 0.7 <s,1024> → 0.9 <f,256> → 0.1 <256,P4> → 0.5 <256,AMD> → 0.5 <f,512> → 0.9 <512,P4> → 0.7 <512,AMD> → 0.5 <f,1024> → 1 <1024,AMD> → 0.5 <1024,P4> → 0.9 val(s,256,P4) = min(0.2,0.6,0.5,0.5) = 0.2 val(f,1024,P4)=min(0.9,1,0.7,0.9,0.7)=0.7 Solutions • Sol(C) = X(C) • an element of A for each tuple of elements of D • val(t) = f1(t|con1) x ... x fk(t|conk)

  18. Multi-criteria problems • Main idea: one semiring for each criteria • Given n c-semirings Si = <Ai, +i, xi, 0i,1i>, we can build the c-semiring <<A1,..., An>, +,x, <01,...,0n>,<11,...,1n>> • + and x obtained by pointwise application of +i and xi on each semiring • A tuple of values associated with each variable instantiation • A partial order even if all the criteria are totally ordered

  19. Example • The problem: choosing a route between two cities • Each piece of highway has a preference and a cost • We want to both minimize the sum of the costs and maximize the preference • Semiring: by putting together one fuzzy semiring and one weighted semiring: • <[0,1],max,min,0,1> • <N, min, +, +, 0> • Best solutions: routes such that there is no other route with a better semiring value • <0.8,$10> is better than <0.7,$15> • Two total orders, but the resulting order is partial: • <0.6, $10> and <0.4,$5> are not comparable

  20. Typical questions • Find an optimal solution • Difficult: NP-hard (branch and bound + constraint propagation) • Is t an optimal solution? • Difficult: NP-hard (we have to find the optimal preference level) • Is t better than t’? • Easy: Linear in the number of constraints (compute the two pref. levels and compare them)

  21. How to find optimal solutions • Classical constraints: • depth-first backtracking search • Constraint propagation • Variable elimination • Local search • ... • Is it possible to extend/adapt these techniques to soft constraints?

  22. Branch and bound • Same scheme as for optimization CSPs • Lower bound = preference of best solution so far (0 at the beginning) • Upper bound for each node • If ub > lb  enter the subtree

  23. Upper bounds • Based on preferences • t = assignment to past variables (P) • ub1= x c(t), with c already assigned by t • ub2 = x ub(cji), where xi in F, xj in P • ub(cji) = +cji(vj,a) for a in Di • ub3 = x (+ cij(a,b)) • x over xi, xj in F • + over a in Di, b in Dj • Ub = ub1 x ub2 x ub3

  24. Upper boundsweighted constraints (min +) • t = assignment to past variables • ub1= + c(t), with c already assigned by t • ub2 = + ub(cji), for xi in F, xj in P • ub(cji) = min cji(vj,a) for a in Di • ub3 = + (min cij(a,b)) • + over xi, xj in F • min over a in Di, b in Dj • ub = ub1 + ub2 + ub3

  25. Upper boundsfuzzy constraints (max min) • t = assignment to past variables • ub1= min c(t), with c already assigned by t • ub2 = min ub(cji), for xi in F, xj in P • ub(cji) = max cji(vj,a) for a in Di • ub3 = min (max cij(a,b)) • min over xi, xj in F • max over a in Di, b in Dj • ub = min(ub1, ub2, ub3)

  26. Constraint propagation • Constraint propagation (ex.arc-consistency): • Deletes an element a from the domain of a variable x if, according to a constraint between x and y, it does not have any compatible element b in the domain of y • Iterate until stability • Polynomial time • Very useful at each node of the search tree to prune subtrees

  27. a b a b X Y a a a b Example No matter what the other constraints are, X=b cannot participate in any solution. So we can delete it without changing the set of solutions.

  28. Properties of GI algorithm • Equivalence: each step preserves the set of solutions • Termination (with finite domains) • Order-independence: no matter how constraints are chosen, when stability is achieved we always have the same problem

  29. Soft constraint propagation • Deleting a value means passing from 1 to 0 in the semiring <{0,1},or,and,0,1> • In general, constraint propagation can change preferences to lower values in the ordering • Soft arc-consistency: given cx, cxy, and cy, computecx := (cx x cxy x cy)|x • Iterate until stability according to algorithm GI

  30. 256  0.5 512  0.8 1024  0.7 s=slow  0.2 f=fast  0.9 VGA MB <s,256> → 0.6 <s,512> → 0.7 <s,1024> → 0.9 <f,256> → 0.1 <f,512> → 0.9 <f,1024> → 1 Example: fuzzy arc-consistency cx := (cx x cxy x cy)|x Fuzzy semiring= <[0,1],max,min,0,1>  +=max an x=min VGA=s  max(min(0.2,0.6,0.5),min(0.2,0.7,0.8),min(0.2,0.9,0.7))= max(0.2,0.2,0.2) = 0.2 VGA=f max(min(0.9,0.1,0.5),min(0.9,0.9,0.8),min(0.9,1,0.7))= max(0.1,0.8,0.7)=0.8

  31. Properties of soft GI • If x idempotent (ex.:fuzzy,classical): • Equivalence • Termination • Order-independence • If x not idempotent (ex.: weighted CSPs, prob.), we could count more than once the same constraint  we need to compensate by subtracting appropriate quantities somewhere else  we need an additional property (fairness=presence of -) • Equivalence • Termination • Not order-independence

  32. Branch and bound with soft constraint propagation • Soft constraint propagation lowers the preferences • ub1, ub2, ub3 can be lower • more probable that ub < lb • more probable to prune

  33. Cuts • Given P soft, and given  in the semiring, build P : • Same graph as P • In each constraint, only the tuples with preference >

  34. 256  0.5 512  0.8 1024  0.7 s=slow  0.2 f=fast  0.9 VGA MB 256  0 512  1 1024  1 s 0 f  1 <s,256> → 1 <s,256> → 0.6 VGA MB <s,512> → 1 <s,512> → 0.7 <s,1024> → 1 <s,1024> →0.9 <f,256> → 0 <f,256> → 0.1 <f,512> → 1 <f,512> → 0.9 <f,1024> → 1 <f,1024> → 1 Example P0.5:

  35. Properties • t optimal solution of P with preference v • For each <v, P consistent and t solution of P • For each  ≥ v, P inconsistent • Hold if x idempotent

  36. Solution method • To find the optimal solutions of P: • Find the consistent P such that, for each > , P inconsistent • The solutions of P are the optimal solutions of P • Just check the P with  occurring in P  O(kd2) calls to a classical constraint solver • k= number of constraints • d = domain size • Binary search  O(log(kd2)) calls

  37. 256  0.5 512  0.8 1024  0.7 s=slow  0.2 f=fast  0.9 VGA MB 256  0 512  1 1024  1 s 0 f  1 <s,256> → 0.6 <s,256> → 1 VGA MB <s,512> → 0.7 <s,512> → 1 <s,1024> → 1 <s,1024> →0.9 <f,256> → 0.1 <f,256> → 0 <f,512> → 1 <f,512> → 0.9 <f,1024> → 1 <f,1024> → 1 Example Optimal solution: VGA=f,MB=512 (val=0.8) P0.5: VGA=f,MB=512 is a solution, but also VGA=f,MB=1024 is a solution

  38. 256  0.5 512  0.8 1024  0.7 s=slow  0.2 f=fast  0.9 VGA MB 256  0 512  1 1024  1 s 0 f  1 <s,256> → 0.6 <s,256> → 0 VGA MB <s,512> → 0.7 <s,512> → 1 <s,1024> → 1 <s,1024> →0.9 <f,256> → 0.1 <f,256> → 0 <f,512> → 1 <f,512> → 0.9 <f,1024> → 1 <f,1024> → 1 Example Optimal solution: VGA=f,MB=512 (val=0.8) P0.6: VGA=f,MB=512 is a solution, but also VGA=f,MB=1024 is a solution

  39. 256  0.5 512  0.8 1024  0.7 s=slow  0.2 f=fast  0.9 VGA MB 256  0 512  1 1024  0 s 0 f  1 <s,256> → 0.6 <s,256> → 0 VGA MB <s,512> → 0.7 <s,512> → 0 <s,1024> → 1 <s,1024> →0.9 <f,256> → 0.1 <f,256> → 0 <f,512> → 1 <f,512> → 0.9 <f,1024> → 1 <f,1024> → 1 Example Optimal solution: VGA=f,MB=512 (val=0.8) P0.7: Only VGA=f,MB=512 is a solution

  40. 256  0.5 512  0.8 1024  0.7 s=slow  0.2 f=fast  0.9 VGA MB 256  0 512  0 1024  0 s 0 f  1 <s,256> → 0.6 <s,256> → 0 VGA MB <s,512> → 0.7 <s,512> → 0 <s,1024> → 1 <s,1024> →0.9 <f,256> → 0.1 <f,256> → 0 <f,512> → 1 <f,512> → 0.9 <f,1024> → 1 <f,1024> → 1 Example Optimal solution: VGA=f,MB=512 (val=0.8) P0.8: No solution

  41. Variable elimination • Choose a variable, x • x is connected via constraints C to the set of variables Y • Compute x(C) and project over Y, and add the new constraint over Y (we need idempotent x) • Forget about x and C • The same for all other variables • When just one variable, its highest preference is the preference of optimal solutions • To find an optimal solution, instantiate the variables in the order opposite to elimination, always taking the value with the highest preference

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