William Lam March 20, 2013 (with slides from the IJCAI-09 tutorial on Combinatorial Optimization for Graphical Models) Discrete Optimization via Branch and Bound with Soft Arc Consistency

Introduction  Bayes nets/Markov random fields/factor graphs  Most Probable Explanation (MPE), Maximum a Posteriori (MAP): maximizing probability  Weighted CSP/Cost function network  Minimum cost  Problems are reducible to each other 2

3 X1X1 X3X3 X5X5 X4X4 X2X2 Select a variable Search Basic Step: Conditioning

4 X1X1 X3X3 X5X5 X4X4 X2X2 X3X3 X5X5 X4X4 X2X2 X3X3 X5X5 X4X4 X2X2 X3X3 X5X5 X4X4 X2X2 …... X 1  a X 1  b X 1  c Search Basic Step: Conditioning

Depth-First Branch and Bound (DFBB) (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 subproblem (defined by current conditioning) LB ffff f  = f  k = k k 5

Computing Bounds  Static Mini-Bucket heuristics (Kask & Dechter, AIJ’01), (Marinescu & Dechter, IJCAI’05)  (with max-marginal matching) (Ihler et al. UAI’12)  Dynamic Mini-Bucket heuristics (Marinescu & Dechter, IJCAI’05)  Maintaining local consistency (Larrosa & Schiex, AAAI’03), (de Givry et al., IJCAI’05)

Local Consistency in Constraint Networks  Massive local inference  Time efficient (local inference, as mini buckets)  Infer only small constraints, added to the network  No variable is eliminated  Produces an equivalent more explicit problem  May detect inconsistency (prune tree search) Arc consistency inference in the scope of 1 constraint 7

 Arc Consistency (binary CSP)  for a constraint c AB and variable A w v v w AB   ABc AB vv0 vw0 wv  ww  AcAcA v0 w  c AB + c B Elim B (min) Applied iteratively on all constraint/variables Empty domain => inconsistency 8

1 Arc Consistency and Cost Functions  for a cost function f AB and a variable A w v v w AB 2 1 ABf AB vv0 vw0 wv2 ww1 AgAgA v0 w1 f AB + f B Elim B (min) EQUIVALENCE LOST 9

1 1 2 Subtract from source in order to preserve the problem  Equivalence Preserving Transformation Shifting Costs (cost compensation) w v v w AB 1 0 ABf AB vv0 vw0 wv ww AgAgA v0 w Elim B 10

Complete Inference vs Local Inference  Local consistency  Combine, eliminate, add & subtract  Massive local inference  Space/time efficient  Preserves equivalence  Provides a lb f  11  Complete inference  Combine, eliminate, add & forget  Systematic inference  Exponential time/space  Preserves optimum  Provides the optimum f 

 Shifting costs from f AB to A Shift(f AB,(A,w),1)  Can be reversed (e.g. Shift(f AB,(A,w), -1)) 0 1 Equivalence Preserving Transformation w v v w AB 1 Arc EPT: shift cost in the scope of 1 cost function Problem structure preserved 12

Equivalence Preserving Transformations EPTs may cycle EPTs may lead to different f 0 Which EPTs should we apply? Shift(f AB,(A,b),1) Shift(f A,(),1) Shift(f AB,(B,a),-1) Shift(f AB,(B,a),1) AB ABAB 13

Node Consistency (NC * )  For any variable A  a, f  + f A (a)<k  a, f A (a)= 0  Complexity: O(nd) w v v v w w f =f = k = A B C Shift(f C, ,1)Shift(f A, ,-1);Shift(f A, ,1) (Larrosa, AAAI 2002) 14

0 0 Arc Consistency (AC * )  NC *  For all f AB  a  b f AB (a,b)= 0  b is a simple-support  complexity: O(n 2 d 3 ) w v v w w f =f = k= A B C Shift(f AC,(C,v),2)Shift(f BC,(B,v),1) Shift(f BC,(B,w),1) Shift(f B, ,1)Shift(f C, ,-2)Shift(f C, ,2) (Larrosa, AAAI 2002) (Schiex, CP 2000) 15

Full Arc Consistency (FAC*)  NC *  For all f AB  a  b f AB (a,b) + f B (b) = 0  b is a full-support  Too strong to enforce

1 1 Directional AC (DAC * )  NC *  For all f AB (A<B)  a  b f AB (a,b) + f B (b) = 0  complexity: O(ed 2 ) w v v v w w f =f = k= A B C A<B<CA<B<C (Cooper, Fuzzy Sets and Systems 2003) Shift(f BC,(C,v),-1) Shift(f BC,(B,v),1)Shift(f BC,(B,w),1) Shift(f B, ,1) Shift(f A, , -2)Shift(f A, , 2) 17

DAC lb = Mini-Bucket(2) lb bucket A: bucket E: bucket D: bucket C: bucket B: f(b,e) f(a,d) hE(Ø)hE(Ø) h B (e) h B (d) h D (a) f(a,b)f(b,d) f(c,e)f(a,c) h C (e) lb = lower bound Mini-buckets A BC D E A < E < D < C < B h B (a) DAC provides an equivalent problem: incrementally DAC+NC (value pruning) can improve lb h C (a) 18

Other Local Consistencies  FDAC* = DAC+AC+NC  Stronger lower bound  O(end 3 )  Better compromise  EDAC* = FDAC+ EAC (existential AC)  Even stronger  O(ed 2 max{nd, k})  Currently among the best practical choice (Larrosa & Schiex, IJCAI 2003) (Cooper, Fuzzy Sets and Systems 2003) (Larrosa & Schiex, AI 2004) (Cooper & Schiex, AI 2004) (de Givry et al., IJCAI 2005) (Sanchez et al, Constraints 2008) 19

FDAC*  DAC* + AC*  For all f AB (A<B)  a  b f AB (a,b) + f B (b) = 0  For all f AB (A>B)  a  b f AB (a,b)= 0 w v w w f =f = k= A B C A<B<CA<B<C 2 v

Existential Arc Consistency (EAC*)  Getting closer to FAC*  For all f AB  a  b f A (a) = 0 f AB (a,b) + f B (b) = 0 (A full support in both directions)  If a variable A is not EAC, we can enforce full- supports and break NC  Leads to increasing lower bound

EDAC*  FDAC* + EAC*  Every value is fully supported in one direction and simply supported in the other (FDAC*)  At least one value per variable is fully supported in both directions (EAC*)  FDAC* < EDAC* < FAC*

FDAC*

EDAC*

EAC* DAC* AC* NC*

Other (Stronger) Consistency Properties  Optimal Soft Arc Consistency  Finds an optimal set of rational EPTs (Cooper et al., IJCAI’07) (Schlesinger, Kibernetika 1976)  Virtual Arc Consistency  Finds an improving sequence of rational EPTs (Cooper et al., AAAI’08)

Higher Arity Functions  Consistency enforcing ignores functions with arity > 2  Must wait until sufficient conditioning reduces function arity to 2  Enhancement: preprocess problem before search  Perform EPTs on functions with arity > 2 to generate binary functions ABCf ABf ACf

Variable/Value Selection  Weighted Degree  For each constraint, count the number of times an inconsistency is reached during search  Variable x i weight is the sum of the weights of all constraints involving x i and another uninstantiated variable (Boussemart et al., ECAI’04)  Last conflicts  Maintain a set of variables that resulted in a dead end and prioritize testing these first (Lecoutre et al., AI’09)  Values: use unary functions as lower bounds

Overall Algorithm  Input: a cost function network N (preprocessed to enforce consistency)  Output: solution cost  Initialize: UB = k solve(N): if all domains are empty, return N 0 X = VariableSelectionHeuristic(N) a = argmin N(X) N’ = EnforceConsistency(N| X=a ) if N’ 0 >= UB, N’ = EnforceConsistency(N| X!=a ) // remove a from N.D(X) return UB = min(UB, solve(N’))

UAI’08 Competition Results 31

UAI’08 Competition Results (II) 32

UAI’08 Competition Results (III) 33

Software  Toulbar (Open source WCSP, MPE solver in C++)  Participated in CP06, CP08, UAI08, UAI10, and PIC11 competitions 34

DAOOPT and Toulbar2  DAOOPT (our AOBB-based solver)  Bounds computed before search via MBE “Top-down approach”: Variable duplication to reduce cluster size Re-parameterization to enforce consistency between copies of variables (FGLP+JGLP+MBE-MM) (Ihler et al., UAI’12) Fast node expansion  Static variable ordering  Toulbar2  Bounds incrementally improved during search via SAC “Bottom-up approach”: Ignores larger factors until conditioning reduces their size Re-parameterization directly generates bounds Node expansion speed depends on consistency algorithm  Dynamic variable ordering 35