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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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 (B i, V i ), B i requested goods, V i value … find the best subset of compatible bids Best = maximize revenue (sum)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-5 Why soft constraints  Multiple sequence alignment (DNA,AA) Given k homologous sequences…  AATAATGTTATTGGTGGATCGATGA  ATGTTGTTCGCGAAGGATCGATAA … find the best alignment (sum)  AATAATGTTATTGGTG---GATCGATGATTA  ----ATGTTGTTCGCGAAGGATCGATAA---

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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.

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-10 Soft constraints  Two new difficulties: How to express preferences in the model? How to solve the resulting optimization problem?

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-11 Solving soft constraints as a CSP  Using a global criteria constraint New constraint S k on all X S k (X) = criteria value must be more than k  Number of variables having value blue  Apply: k := n Repeat solve the original problem  S k (X) k:=k-1; Until solution n-ary constraint Inefficient!!

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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 

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-13 Soft constraint network (X,D,C)(X,D,C) X={x 1,..., x n } variables D={D 1,..., D n } finite domains C={c 1,..., c e } cost functions  var(c i ) scope  c i (t):  E (ordered cost domain, T,  )  Obj. Function: F(X)=  c i (X)  Solution: F(t)  T  Soft CN: find minimal cost solution identity annihilator commutative associative monotonic

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-14 Semiring CSP Valued CSP and Semiring CSP Valued CSP (total order)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-15 Specific frameworks  E = {t,f}  = andclassical CSP  E = [0,1]  = max  fuzzy CSP  E = N  = +weighted CSP  E = [0,1]  = *bayesian net Lexicographic CSP, probabilistic CSP…

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-16 Weighted CSP example (  = +) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): number of non blue vertices For each vertex For each edge: xixi c(x i ) b0 g1 r1 xixi xjxj c(x i,x j ) bbT bg0 br0 gb0 ggT gr0 rb0 rg0 rrT

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-17 Fuzzy constraint network (  =max) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): highest color used (b<g<r<y) For each vertex Connected vertices must have different colors xixi c(x i ) b0.2 g0.4 r0.6 y0.8

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-19 Weighted CSP example (  = +) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): number of non blue vertices For each vertex For each edge: xixi c(x i ) b0 g1 r1 xixi xjxj c(x i,x j ) bbT bg0 br0 gb0 ggT gr0 rb0 rg0 rrT T=6 c  = 0 T=3 Optimal coloration with less than 3 non-blue

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-20 Microstructural graph b g r x5x5 x4x4 x2x2 x1x1 x3x3 T=6 c  =

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-21 WCSP: Example w v v v w w C  = 0 T= x y z X={x y z} D i ={v w} C={C xz C yz C x C y C z C  } Valuation: 2  1  2  1  0  0 = T Not a solution Valuation: 0  1  0  1  0  0 = 2 (optimal) solution

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-22 Basic operations on cost functions I.Assignment (conditioning) II.Combination (join) III.Projection (elimination)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-23 Assignment (conditioning) xixi xjxj c(x i,x j ) bbT=6 bg0 br0 gb0 gg gr0 rb0 rg0 rr Assign(c,x i,b) xjxj bT=6 g0 r0 f(x j ) Assign(f,x j,g) 0 hh

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-24 Combination (join with  ) xixi xjxj c(x i,x j ) bb6 bg0 gb0 gg6 xjxj xkxk c(x j,x k ) bb6 bg0 gb0 gg6 xixi xjxj xkxk f(x i,x j,x k ) bbb12 bbg6 bgb0 bgg6 gbb6 gbg0 ggb6 ggg  = 0  6

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-25 Projection (elimination) xixi xjxj c(x i,x j ) bb4 bg6 br0 gb2 gg6 gr3 rb1 rg0 rr6 Elim(c,x j ) xixi f(x i ) b g r Elim(f,x i ) hh 0 Min

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-26 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-27 Systematic search (LB) Lower Bound (UB) Upper Bound If  UB then prune variables under estimation of the best solution in the sub-tree = best solution so far Each node is a soft constraint subproblem cc LB = c  = T

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-28 Assigning x 3 x3x3 x2x2 x5x5 x1x1 x4x4 x2x2 x5x5 x1x1 x4x4 x2x2 x5x5 x1x1 x4x4 x2x2 x5x5 x1x1 x4x4

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-29 T Assign g to X 0 x5x5 x4x4 x2x2 x1x1 x3x3 T=6 c=c= b g r T T b g r 34 T Backtrack

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-30 Depth First Search (DFS) BT(X,D,C) if (X=  ) then Top :=c  else x j := selectVar(X) forall a  D j do  c  C s.t. xj  var(c) c:=Assign(c, x j,a) c  :=  c  C s.t. var(c)=  c if (LB<Top) then BT(X-{x j },D-{D j },C) Heuristics Improve LB Good UB ASAP

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-31 Improving the LB  Assigned constraints (c  )  Original constraints Solve Optimal cost  c  Gives a stronger LB Can be solved beforehand

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-32 Russian Doll Search  static variable ordering  solves increasingly large subproblems  uses previous LB recursively  May speedup search by several orders of magnitude X1X1 X2X2 X3X3 X4X4 X5X5

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-33 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-34 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-35 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  3 hours UB initialized to 3230  1 hour  Local search can find the optimum in a few minutes

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-36 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-37 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-38 CC Variable elimination  Select a variable X i  Compute the set K i of constraints that involves this variable  Add Elim( )  Remove variable and K i  Time:  (exp(deg i +1))  Space:  (exp(deg i )) X 4 X 3 X5X5 X 2 X 1

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-39 X 5 Graph induced by an elimination order  The original graph plus all the extra edges for computed constraints  Induced width = max number of variables involved in a K i  = max number of neighbors after. X 4 X 3 X 2 X Induced width = 2

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-40 Variable elimination  Time: exponential in the size of the largest combination performed (1+induced width)  Space: similar !  Infeasible as a general method…

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-41 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 like cycle-cutset

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-42 Example BB-VE(2)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-43 Example BB-VE(2) cc cc cc

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-44 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-45 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-46 Local consistency  Local property enforceable in polynomial time yielding an equivalent and more explicit problem  Massive local (incomplete) inference  Very useful in classical CSP Node consistency Arc consistency…

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-47 Classical arc consistency (binary CSP)  for any X i and c ij f=Elim(c ij  c i  c j,X j ) brings no new information on X i w v v w ij T T XiXi XjXj c ij vv0 vw0 wvT wwT XiXi f(X i ) v0 wT c ij  c i  c j Elim x j T

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-48 Arc consistency and soft constraints  for any X i and c ij f=Elim(c ij  c i  c j,X j ) brings no new information on X i w v v w ij 2 1 XiXi XjXj c ij vv0 vw0 wv2 ww1 XiXi f(X i ) v0 w1 c ij  c i  c j Elim x j 1 EQUIVALENCE LOST … unless  idempotent (fuzzy CSP)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-49 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 xixi xjxj c(x i,x j ) bb0.6 bg0.3 gb0.1 gg0.5 Cut(c, 0.4) xixi xjxj c(x i,x j ) bbT bg0 gb0 ggT

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-50 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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-51 Back to Weighted CSP and Arc consistency  for any X i and c ij f=Elim(c ij  c i  c j,X j ) brings no new information on X i w v v w ij 2 1 XiXi XjXj c ij vv0 vw0 wv2 ww1 XiXi f(X i ) v0 w1 c ij  c i  c j Elim x j 1 EQUIVALENCE LOST … unless  idempotent (fuzzy CSP)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-52  Projection of c ij on X i with compensation  Equivalence preserving transformation  Can be reversed  Requires the existence of a pseudo difference (a b)  b = a 1 A new operation on soft networks w v v w ij 1 0

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-53 Node Consistency (NC * ) For all variable i   a, C  + C i (a)<T   a, C i (a)= 0 Complexity: O(nd) w v v v w w C =C = T = x y z

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-54 0 Arc Consistency (AC * ) NC * For all C ij   a  b C ij (a,b)= 0 b is a support complexity: O(n 2 d 3 ) w v v w w C =C = T= x y z

August 24, 2004ECAI 2004 Tutorial: Constraint processingB Directional AC (DAC * ) NC * For all C ij (i<j)   a  b C ij (a,b) + C j (b) = 0 b is a full-support complexity: O(ed 2 ) w v v v w w C =C = T= x y z x<y<zx<y<z

August 24, 2004ECAI 2004 Tutorial: Constraint processingB Full AC (FAC * ) NC * For all C ij   a  b C ij (a,b) + C j (b) = 0 (full support) w v v w C  =0 T= x z 1 1 That’s our starting point! No termination !!!

August 24, 2004ECAI 2004 Tutorial: Constraint processingB Full DAC (FDAC * ) NC * For all C ij (i<j)   a  b C ij (a,b) + C j (b) = 0 (full support) For all C ij (i>j)   a  b, C ij (a,b) = 0 (support) Complexity: O(end 3 ) w v v v w w C =C = T= x y z x<y<zx<y<z

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-58 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)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-59 Boosting search with LC BT(X,D,C) if (X=  ) then Top :=c  else x j := selectVar(X) forall a  D j do  c  C s.t. xj  var(c) c:=Assgn(c, x j,a) c  :=  c  C s.t. var(c)=  c if (LC) then BT(X-{x j },D-{D j },C)

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-60 BT MNC MAC/MDAC MFDAC

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-61 Boosting Systematic Search with Local search  Ex: Frequency assignment problem Instance: CELAR6-sub4  #var: 22, #val: 44, Optimum: 3230 Solver: toolbar MNC*  1 year (estimated) MFDAC*  1 hour

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-62 CPU time n. of variables

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-63  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

August 24, 2004ECAI 2004 Tutorial: Constraint processingB-64 Known implementations  Con'Flex: fuzzy CSP system with integer, symbolic and numerical constraints ( ).  clp(FD,S): semi-ring CLP ( pauillac.inria.fr/.georget/clp_fds/clp_fds.html ).  LVCSP: Common-Lisp library for Valued CSP with an emphasis on strictly monotonic operators ( ftp.cert.fr/pub/lemaitre/LVCSP ).  Choco: a claire library for CSP. Existing layers above Choco implements some WCSP algorithms ( ).  toolbar: C library for MaxCSP and related problems ( carlit.toulouse.inra.fr/cgi-bin/awki.cgi/ToolBarIntro ). carlit.toulouse.inra.fr/cgi-bin/awki.cgi/SoftCSP