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Soft constraint processing

Soft constraint processing. Thomas Schiex INRA – Toulouse France Pedro Meseguer CSIC – IIIA – Barcelona Spain. Special thanks to: Javier Larrosa UPC – Barcelona Spain. Overview. Introduction and definitions Why soft constraints and dedicated algorithms? Generic and specific models

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Soft constraint processing

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  1. Soft constraint processing Thomas Schiex INRA – Toulouse France Pedro Meseguer CSIC – IIIA – Barcelona Spain Special thanks to: Javier Larrosa UPC – Barcelona Spain

  2. Overview • Introduction and definitions • Why soft constraints and dedicated algorithms? • Generic and specific models • Solving soft constraint problems • By search (systematic and local) • By complete inference… and search • By incomplete inference… and search • Existing implementations ECAI 2004 Tutorial: Constraint processing

  3. Why soft constraints? • CSP framework: for decision problems • Many problems are optimization problems • Economics (combinatorial auctions) • Given a set G of goods and a set B of bids… • Bid (Bi , Vi ), Bi requested goods, Vi value • … find the best subset of compatible bids • Best = maximize revenue (sum) ECAI 2004 Tutorial: Constraint processing

  4. Why soft constraints? • Satellite scheduling • Spot 5 is an earth observation satellite • It has 3 on-board cameras • Given a set of requested pictures (of different importance)… • Resources, data-bus bandwidth, setup-times, orbiting • …select a subset of compatible pictures with max. importance (sum) ECAI 2004 Tutorial: Constraint processing

  5. Why soft constraints • Multiple sequence alignment (DNA,AA) • Given k homologous sequences… • AATAATGTTATTGGTGGATCGATGA • ATGTTGTTCGCGAAGGATCGATAA • … find the best alignment (sum) • AATAATGTTATTGGTG---GATCGATGATTA • ----ATGTTGTTCGCGAAGGATCGATAA--- ECAI 2004 Tutorial: Constraint processing

  6. 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) ECAI 2004 Tutorial: Constraint processing

  7. Why soft constraints? • Ressource allocation (frequency assignment) • Given a telecommunication network • …find the best frequency for each communication link • Best can be: • Minimize the maximum frequency (max) • Minimize the global interference (sum) ECAI 2004 Tutorial: Constraint processing

  8. Why soft constraints • Even in decision problems, the user may have preferences among solutions. • It happens in most real problems. Experiment: give users a few solutions and they will find reasons to prefer some of them. ECAI 2004 Tutorial: Constraint processing

  9. Overview • Introduction and definitions • Why soft constraints and dedicated algorithms? • Generic and specific models • Solving soft constraint problems • By search (systematic and local) • By complete inference… and search • By incomplete inference… and search • Existing implementations ECAI 2004 Tutorial: Constraint processing

  10. Soft constraints • Two new difficulties: • How to express preferences in the model? • How to solve the resulting optimization problem? ECAI 2004 Tutorial: Constraint processing

  11. Solving soft constraints as a CSP • Using a global criteria constraint • New constraint Sk on all X • Sk(X) = criteria value must be more than k • Number of variables having value blue • Apply: n-ary constraint Inefficient!! k := n Repeat solve the original problem  Sk(X) k:=k-1; Until solution ECAI 2004 Tutorial: Constraint processing

  12. Combined local preferences • Constraints are local cost functions • Costs combine with a dedicated operator • max: priorities • +: additive costs • *: factorized probabilities… • Goal: find an assignment with the optimal combined cost  Fuzzy/min-max CSP Weighted CSP Probabilistic CSP, BN ECAI 2004 Tutorial: Constraint processing

  13. Soft constraint network • (X,D,C) • X={x1,..., xn} variables • D={D1,..., Dn} finite domains • C={c1,..., ce} cost functions • var(ci) scope • ci(t): E (ordered cost domain, T, ) • Obj. Function: F(X)= ci (X) • Solution: F(t)  T • Soft CN: find minimal cost solution annihilator identity • commutative • associative • monotonic ECAI 2004 Tutorial: Constraint processing

  14. Valued CSP and Semiring CSP Semiring CSP Valued CSP (total order) ECAI 2004 Tutorial: Constraint processing

  15. Specific frameworks • E = {t,f}  = and classical CSP • E = [0,1]  = max  fuzzy CSP • E = N   = + weighted CSP • E = [0,1]  = * bayesian net Lexicographic CSP, probabilistic CSP… ECAI 2004 Tutorial: Constraint processing

  16. Weighted CSP example ( = +) For each vertex x3 x1 x2 x4 For each edge: x5 F(X): number of non blue vertices ECAI 2004 Tutorial: Constraint processing

  17. Fuzzy constraint network (=max) For each vertex x3 x1 x2 x4 Connected vertices must have different colors x5 F(X): highest color used (b<g<r<y) ECAI 2004 Tutorial: Constraint processing

  18. Some insight in unusual items • T = maximum violation. • Can be set to a bounded max. violation k • Solution: F(t) < k = Top • Empty scope soft constraint c (a constant) • Gives an obvious lower bound on the optimum • If you do not like it: c =  Additional expression power ECAI 2004 Tutorial: Constraint processing

  19. Weighted CSP example ( = +) For each vertex T=6 T=3 x3 c = 0 x1 x2 x4 For each edge: x5 F(X): number of non blue vertices Optimal coloration with less than 3 non-blue ECAI 2004 Tutorial: Constraint processing

  20. Microstructural graph b g r x3 0 1 1 6 6 6 x1 x4 0 1 1 0 1 1 0 1 1 x2 T=6 c=0 x5 0 1 1 ECAI 2004 Tutorial: Constraint processing

  21. Basic operations on cost functions • Assignment (conditioning) • Combination (join) • Projection (elimination) ECAI 2004 Tutorial: Constraint processing

  22. Assignment (conditioning) Assign(c,xi,b) f(xj) Assign(f,xj,g) h ECAI 2004 Tutorial: Constraint processing

  23. Combination (join with )  = 0  6 ECAI 2004 Tutorial: Constraint processing

  24. Projection (elimination) Min Elim(c,xj) Elim(f,xi) 0 0 2 0 ECAI 2004 Tutorial: Constraint processing

  25. Overview • Introduction and definitions • Why soft constraints and dedicated algorithms? • Generic and specific models • Solving soft constraint problems • By search (systematic and local) • By complete inference… and search • By incomplete inference… and search • Existing implementations ECAI 2004 Tutorial: Constraint processing

  26. Systematic search Each node is a soft constraint subproblem variables (LB) Lower Bound = c under estimation of the best solution in the sub-tree c If  UB then prune LB = T (UB) Upper Bound = best solution so far ECAI 2004 Tutorial: Constraint processing

  27. Assigning x3 x3 x1 x2 x4 x5 x1 x2 x4 x1 x2 x4 x1 x2 x4 x5 x5 x5 ECAI 2004 Tutorial: Constraint processing

  28. Assign g to X 4 3 g b r x3 0 1 1 6 6 g b r 6 x1 x4 0 T 1 1 0 T 1 1 x2 0 1 1 T T=6 T 1 0 c= Backtrack x5 0 1 1 ECAI 2004 Tutorial: Constraint processing

  29. Depth First Search (DFS) BT(X,D,C) if (X=) then Top :=c else xj := selectVar(X) forallaDjdo cC s.t. xjvar(c)c:=Assign(c, xj ,a) c:= cC s.t. var(c)= c if (LB<Top) then BT(X-{xj},D-{Dj},C) Heuristics Improve LB Good UB ASAP ECAI 2004 Tutorial: Constraint processing

  30. Improving the LB • Assigned constraints (c) • Original constraints Solve Optimal cost  c • Gives a stronger LB • Can be solved beforehand ECAI 2004 Tutorial: Constraint processing

  31. Russian Doll Search • static variable ordering • solves increasingly large subproblems • uses previous LB recursively • May speedup search by several orders of magnitude X1 X2 X3 X4 X5 ECAI 2004 Tutorial: Constraint processing

  32. 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 ECAI 2004 Tutorial: Constraint processing

  33. Boosting Systematic Search with Local Search • 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 Local search (X,D,C) Sub-optimal solution time limit ECAI 2004 Tutorial: Constraint processing

  34. Boosting Systematic Search with Local Search • Ex: Frequency assignment problem • Instance: CELAR6-sub4 • #var: 22 , #val: 44 , Optimum: 3230 • Solver: toolbar with default options • UB initialized to 100000  3 hours • UB initialized to 3230  1 hour • Local search can find the optimum in a few minutes ECAI 2004 Tutorial: Constraint processing

  35. Overview • Introduction and definitions • Why soft constraints and dedicated algorithms? • Generic and specific models • Solving soft constraint problems • By search (systematic and local) • By complete inference… and search • By incomplete inference… and search • Existing implementations ECAI 2004 Tutorial: Constraint processing

  36. Variable elimination (aka bucket elimination) • Solves the problem by a sequence of problem transformations • Each step yields an equivalent problem with one less variable • When all variables have been eliminated, the problem is solved • Optimal solutions of the original problem can be recomputed ECAI 2004 Tutorial: Constraint processing

  37. Variable elimination • Select a variable Xi • Compute the set Ki of constraints that involves this variable • Add Elim( ) • Remove variable and Ki • Time: (exp(degi+1)) • Space: (exp(degi)) X1 X5 C X4 X2 X3 ECAI 2004 Tutorial: Constraint processing

  38. Variable elimination • Time: exponential in the size of the largest combination performed (1+induced width) • Space: similar ! • Infeasible as a general method… ECAI 2004 Tutorial: Constraint processing

  39. Boosting search with variable elimination: BB-VE(k) • At each node • Select an unassigned variable Xi • If degi ≤ k then eliminate Xi • Else branch on the values of Xi • Properties • BE-VE(-1) is BB • BE-VE(w*) is VE • BE-VE(1) is like cycle-cutset ECAI 2004 Tutorial: Constraint processing

  40. Example BB-VE(2) ECAI 2004 Tutorial: Constraint processing

  41. Example BB-VE(2) c c c ECAI 2004 Tutorial: Constraint processing

  42. 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 ECAI 2004 Tutorial: Constraint processing

  43. Overview • Introduction and definitions • Why soft constraints and dedicated algorithms? • Generic and specific models • Solving soft constraint problems • By search (systematic and local) • By complete inference… and search • By incomplete inference… and search • Existing implementations ECAI 2004 Tutorial: Constraint processing

  44. Local consistency • Local property enforceable in polynomialtime yielding an equivalent and more explicit problem • Massive local (incomplete) inference • Very useful in classical CSP • Node consistency • Arc consistency… ECAI 2004 Tutorial: Constraint processing

  45. Classical arc consistency (binary CSP) • for any Xi and cij • f=Elim(cij ci cj,Xj) brings no new information on Xi cij ci cj v v T 0 0 Elim xj T 0 w w 0 T i j ECAI 2004 Tutorial: Constraint processing

  46. Arc consistency and soft constraints • for any Xi and cij • f=Elim(cij ci cj,Xj) brings no new information on Xi cij ci cj v v 2 0 0 Elim xj 1 0 w w 0 1 i j EQUIVALENCE LOST …unless idempotent (fuzzy CSP) ECAI 2004 Tutorial: Constraint processing

  47. A parenthesis…Why min-max (fuzzy) CSP is easy • A new operation on soft constraints: • The -cut of a soft constraint c is the hard constraint c that forbids t iff c(t) >  • Can be applied to a whole soft CN Cut(c, 0.4) ECAI 2004 Tutorial: Constraint processing

  48. Solving a min-max CSP by slicing • Let S be the set of all optimal solutions of a min-max CN • Let  be the maximum  s.t. the -cut is consistent. Then S is the set of solutions of the -cut. • A min-max (or fuzzy) CN can be solved in O(log(# different costs)) CSP. • Can be used to lift polynomial classes Binary search ECAI 2004 Tutorial: Constraint processing

  49. Back to Weighted CSP andArc consistency • for any Xi and cij • f=Elim(cij ci cj,Xj) brings no new information on Xi cij ci cj v v 2 0 0 Elim xj 1 0 w w 0 1 i j EQUIVALENCE LOST …unless idempotent (fuzzy CSP) ECAI 2004 Tutorial: Constraint processing

  50. A new operation on soft networks • Projection of cij on Xi with compensation • Equivalence preserving transformation • Can be reversed • Requires the existence of a pseudo difference (a b)  b = a v v 1 0 0 0 0 w w 1 i j ECAI 2004 Tutorial: Constraint processing

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