chapter 13
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Chapter 13
Constraint OptimizationAnd counting, and
enumeration275 class
Outline Introduction
Optimization tasks for graphical models Solving optimization problems with inference and
search Inference
Bucket elimination, dynamic programming Mini-bucket elimination
Search Branch and bound and best-first Lower-bounding heuristics AND/OR search spaces
Hybrids of search and inference Cutset decomposition Super-bucket scheme
A Bred greenred yellowgreen redgreen yellowyellow greenyellow red
Example: map coloring Variables - countries (A,B,C,etc.)
Values - colors (e.g., red, green, yellow)
Constraints: etc. ,ED D, AB,A
C
A
B
DE
F
G
Task: consistency?Find a solution, all solutions, counting
Constraint Satisfaction
= {(¬C), (A v B v C), (¬A v B v E), (¬B v C v D)}.
Propositional Satisfiability
Constraint Optimization Problemsfor Graphical Models
functionscost - },...,{
domains - },...,{
variables- },...,{
:where,, triplea is A
1
1
1
m
n
n
ffF
DDD
XXX
FDXRCOPfinite
A B D Cost1 2 3 31 3 2 22 1 3 02 3 1 03 1 2 53 2 1 0
m
i i XfXF1
)(
FunctionCost Global
f(A,B,D) has scope {A,B,D}
Constraint Optimization Problemsfor Graphical Models
functionscost - },...,{
domains - },...,{
variables- },...,{
:where,, triplea is A
1
1
1
m
n
n
ffF
DDD
XXX
FDXRCOPfinite
A B D Cost1 2 3 31 3 2 22 1 3 02 3 1 03 1 2 53 2 1 0
G
A
B C
D F
m
i i XfXF1
)(
FunctionCost Global
Primal graph =Variables --> nodesFunctions, Constraints - arcs
f1(A,B,D)f2(D,F,G)f3(B,C,F)
f(A,B,D) has scope {A,B,D}
F(a,b,c,d,f,g)= f1(a,b,d)+f2(d,f,g)+f3(b,c,f)
Constrained Optimization
Example: power plant scheduling
)X,...,ost(XTotalFuelC minimize :
)(Power : demandpower time,down-min and up-min ,, :sConstraint
. domain ,Variables
N1
4321
1
Objective
DemandXXXXX
{ON,OFF}},...,X{X
i
n
Probabilistic Networks
Smoking
BronchitisCancer
X-Ray
Dyspnoea
P(S)
P(B|S)
P(D|C,B)
P(C|S)
P(X|C,S)
P(S,C,B,X,D) = P(S)· P(C|S)· P(B|S)· P(X|C,S)· P(D|C,B)
C BD=0
D=1
0 0 0.1 0.9
0 1 0.7 0.3
1 0 0.8 0.2
1 1 0.9 0.1
P(D|C,B)
Outline Introduction
Optimization tasks for graphical models Solving by inference and search
Inference Bucket elimination, dynamic programming,
tree-clustering, bucket-elimination Mini-bucket elimination, belief propagation
Search Branch and bound and best-first Lower-bounding heuristics AND/OR search spaces
Hybrids of search and inference Cutset decomposition Super-bucket scheme
Computing MPE
“Moral” graph
A
D E
CB
bcde ,,,0max
P(a)P(b|a)P(c|a)P(d|b,a)P(e|b,c)=
0max
eP(a)
dmax
),,,( ecdahB
bmax P(b|a)P(d|b,a)P(e|b,c)
B C
ED
Variable Elimination
P(c|a)c
max
MPE=
b
maxElimination operator
MPE
bucket B:
P(a)
P(c|a)
P(b|a) P(d|b,a) P(e|b,c)
bucket C:
bucket D:
bucket E:
bucket A:
e=0
B
C
D
E
A
e)(a,hD
(a)hE
e)c,d,(a,hB
e)d,(a,hC
Finding )xP(maxMPEx
),|(),|()|()|()(max
,,,,cbePbadPabPacPaPMPE
bcdea
Algorithm elim-mpe (Dechter 1996)Non-serial Dynamic Programming (Bertele and Briochi, 1973)
Generating the MPE-tuple
C:
E:
P(b|a) P(d|b,a) P(e|b,c)B:
D:
A: P(a)
P(c|a)
e=0 e)(a,hD
(a)hE
e)c,d,(a,hB
e)d,(a,hC
(a)hP(a)max arga' 1. E
a
0e' 2.
)e'd,,(a'hmax argd' 3. C
d
)e'c,,d',(a'h
)a'|P(cmax argc' 4.B
c
)c'b,|P(e')a'b,|P(d')a'|P(bmax argb' 5.
b
)e',d',c',b',(a' Return
b
maxElimination operator
MPE
exp(W*=4)”induced width” (max clique size)
bucket B:
P(a)
P(c|a)
P(b|a) P(d|b,a) P(e|b,c)
bucket C:
bucket D:
bucket E:
bucket A:
e=0
B
C
D
E
A
e)(a,hD
(a)hE
e)c,d,(a,hB
e)d,(a,hC
Complexity
),|(),|()|()|()(max
,,,,cbePbadPabPacPaPMPE
bcdea
Algorithm elim-mpe (Dechter 1996)Non-serial Dynamic Programming (Bertele and Briochi, 1973)
Complexity of bucket elimination
))((exp ( * dwrOddw ordering alonggraph primal theof width induced the)(*
The effect of the ordering:
4)( 1* dw 2)( 2
* dwconstraint graph
A
D E
CB
B
C
D
E
A
E
D
C
B
A
Finding smallest induced-width is hard
r = number of functions
Bucket-elimination is time and space
Directional i-consistency
DCBR
A
E
CD
B
D
CB
E
D
CB
E
DC
B
E
:A
B A:B
BC :C
AD C,D :D
BE C,E D,E :E
Adaptive d-arcd-path
DBDC RR ,CBR
DRCRDR
Mini-bucket approximation: MPE task
Split a bucket into mini-buckets =>bound complexity
XX gh )()()O(e :decrease complexity lExponentia n rnr eOeO
Mini-Bucket Elimination
A
B C
D
E
P(A)
P(B|A) P(C|A)
P(E|B,C)
P(D|A,B)
Bucket B
Bucket C
Bucket D
Bucket E
Bucket A
P(B|A) P(D|A,B)P(E|B,C)
P(C|A)
E = 0
P(A)
maxB∏
hB (A,D)
MPE* is an upper bound on MPE --UGenerating a solution yields a lower bound--L
maxB∏
hD (A)
hC (A,E)
hB (C,E)
hE (A)
MBE-MPE(i) Algorithm Approx-MPE (Dechter&Rish 1997)
Input: i – max number of variables allowed in a mini-bucket Output: [lower bound (cost of a sub-optimal solution), upper bound]
Example: approx-mpe(3) versus elim-mpe
2* w 4* w
Properties of MBE(i)
Complexity: O(r exp(i)) time and O(exp(i)) space. Yields an upper-bound and a lower-bound.
Accuracy: determined by upper/lower (U/L) bound.
As i increases, both accuracy and complexity increase.
Possible use of mini-bucket approximations: As anytime algorithms As heuristics in search
Other tasks: similar mini-bucket approximations for: belief updating, MAP and MEU (Dechter and Rish, 1997)
Outline Introduction
Optimization tasks for graphical models Solving by inference and search
Inference Bucket elimination, dynamic programming Mini-bucket elimination
Search Branch and bound and best-first Lower-bounding heuristics AND/OR search spaces
Hybrids of search and inference Cutset decomposition Super-bucket scheme
The Search Space
9
1
mini
iX ff
A
E
C
B
F
D
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
C
D
F
E
B
A 0 1
Objective function:
A B f1
0 0 20 1 01 0 11 1 4
A C f2
0 0 30 1 01 0 01 1 1
A E f3
0 0 00 1 31 0 21 1 0
A F f4
0 0 20 1 01 0 01 1 2
B C f5
0 0 00 1 11 0 21 1 4
B D f6
0 0 40 1 21 0 11 1 0
B E f7
0 0 30 1 21 0 11 1 0
C D f8
0 0 10 1 41 0 01 1 0
E F f9
0 0 10 1 01 0 01 1 2
The Search Space
Arc-cost is calculated based on cost components.
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
C
D
F
E
B
A 0 1
A B f1
0 0 20 1 01 0 11 1 4
A
E
C
B
F
D
A C f2
0 0 30 1 01 0 01 1 1
A E f3
0 0 00 1 31 0 21 1 0
A F f4
0 0 20 1 01 0 01 1 2
B C f5
0 0 00 1 11 0 21 1 4
B D f6
0 0 40 1 21 0 11 1 0
B E f7
0 0 30 1 21 0 11 1 0
C D f8
0 0 10 1 41 0 01 1 0
E F f9
0 0 10 1 01 0 01 1 2
3 02 23 02 23 02 23 02 2 3 02 23 02 23 02 23 02 2
0 0
3 5 3 5 3 5 3 5 1 3 1 3 1 3 1 3
5 6 4 2 2 4 1 0
3 1
2
5 4
0
1 20 41 20 41 20 41 20 4 1 20 41 20 41 20 41 20 4
5 2 5 2 5 2 5 2 3 0 3 0 3 0 3 0
5 6 4 2 2 4 1 0
0 2 2 5
0 4
9
1
mini
iX ff
The Value Function
Value of node = minimal cost solution below it
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
C
D
F
E
B
A 0 1
A B f1
0 0 20 1 01 0 11 1 4
A
E
C
B
F
D
A C f2
0 0 30 1 01 0 01 1 1
A E f3
0 0 00 1 31 0 21 1 0
A F f4
0 0 20 1 01 0 01 1 2
B C f5
0 0 00 1 11 0 21 1 4
B D f6
0 0 40 1 21 0 11 1 0
B E f7
0 0 30 1 21 0 11 1 0
C D f8
0 0 10 1 41 0 01 1 0
E F f9
0 0 10 1 01 0 01 1 2
3 00
2 2
6
2
3
3 02 23 02 23 02 2 3 02 23 02 23 02 23 02 20 0 02 2 2 0 2 0 0 02 2 2
3 3 3 1 1 1 1
8 5 3 1
5
5
1 0 1 1 10 0 0 1 0 1 1 10 0 0
2 2 2 2 0 0 0 0
7 4 2 0
7 4
7
50 0
3 5 3 5 3 5 3 5 1 3 1 3 1 3 1 3
5 6 4 2 2 4 1 0
3 1
2
5 4
0
1 20 41 20 41 20 41 20 4 1 20 41 20 41 20 41 20 4
5 2 5 2 5 2 5 2 3 0 3 0 3 0 3 0
5 6 4 2 2 4 1 0
0 2 2 5
0 4
9
1
mini
iX ff
An Optimal Solution
Value of node = minimal cost solution below it
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
C
D
F
E
B
A 0 1
A B f1
0 0 20 1 01 0 11 1 4
A
E
C
B
F
D
A C f2
0 0 30 1 01 0 01 1 1
A E f3
0 0 00 1 31 0 21 1 0
A F f4
0 0 20 1 01 0 01 1 2
B C f5
0 0 00 1 11 0 21 1 4
B D f6
0 0 40 1 21 0 11 1 0
B E f7
0 0 30 1 21 0 11 1 0
C D f8
0 0 10 1 41 0 01 1 0
E F f9
0 0 10 1 01 0 01 1 2
3 00
2 2
6
2
3
3 02 23 02 23 02 2 3 02 23 02 23 02 23 02 20 0 02 2 2 0 2 0 0 02 2 2
3 3 3 1 1 1 1
8 5 3 1
5
5
1 0 1 1 10 0 0 1 0 1 1 10 0 0
2 2 2 2 0 0 0 0
7 4 2 0
7 4
7
50 0
3 5 3 5 3 5 3 5 1 3 1 3 1 3 1 3
5 6 4 2 2 4 1 0
3 1
2
5 4
0
1 20 41 20 41 20 41 20 4 1 20 41 20 41 20 41 20 4
5 2 5 2 5 2 5 2 3 0 3 0 3 0 3 0
5 6 4 2 2 4 1 0
0 2 2 5
0 4
Basic Heuristic Search Schemes
Heuristic function f(x) computes a lower bound on the best
extension of x and can be used to guide a heuristic search algorithm. We focus on
1.Branch and BoundUse heuristic function f(xp) to prune the depth-first search tree.Linear space
2.Best-First SearchAlways expand the node with the highest heuristic value f(xp).Needs lots of memory
f L
L
Classic Branch-and-Bound
nn
g(n)g(n)
h(n)h(n)
LB(n) = g(n) + h(n)LB(n) = g(n) + h(n)
Lower Bound Lower Bound LBLB
OR Search Tree
Prune if LB(n) ≥ UBPrune if LB(n) ≥ UB
Upper Bound Upper Bound UBUB
How to Generate Heuristics
The principle of relaxed models
Linear optimization for integer programs
Mini-bucket elimination Bounded directional consistency ideas
Generating Heuristic for graphical models(Kask and Dechter, 1999)
Given a cost function
C(a,b,c,d,e) = f(a) • f(b,a) • f(c,a) • f(e,b,c) • P(d,b,a)
Define an evaluation function over a partial assignment as theprobability of it’s best extension
f*(a,e,d) = minb,c f(a,b,c,d,e) = = f(a) • minb,c f(b,a) • P(c,a) • P(e,b,c) • P(d,a,b)
= g(a,e,d) • H*(a,e,d)
D
E
E
DA
D
BD
B0
1
1
0
1
0
Generating Heuristics (cont.)
H*(a,e,d) = minb,c f(b,a) • f(c,a) • f(e,b,c) • P(d,a,b)
= minc [f(c,a) • minb [f(e,b,c) • f(b,a) • f(d,a,b)]]
minc [f(c,a) • minb f(e,b,c) • minb [f(b,a) • f(d,a,b)]]
minb [f(b,a) • f(d,a,b)] • minc [f(c,a) • minb f(e,b,c)]
= hB(d,a) • hC(e,a)
= H(a,e,d)
f(a,e,d) = g(a,e,d) • H(a,e,d) f*(a,e,d)
The heuristic function H is what is compiled during the preprocessing stage of the
Mini-Bucket algorithm.
Generating Heuristics (cont.)
H*(a,e,d) = minb,c f(b,a) • f(c,a) • f(e,b,c) • P(d,a,b)
= minc [f(c,a) • minb [f(e,b,c) • f(b,a) • f(d,a,b)]]
minc [f(c,a) • minb f(e,b,c) • minb [f(b,a) • f(d,a,b)]]
minb [f(b,a) • f(d,a,b)] • minc [f(c,a) • minb f(e,b,c)]
= hB(d,a) • hC(e,a)
= H(a,e,d)
f(a,e,d) = g(a,e,d) • H(a,e,d) f*(a,e,d)
The heuristic function H is what is compiled during the preprocessing stage of the
Mini-Bucket algorithm.
Static MBE Heuristics Given a partial assignment xp, estimate the cost of the
best extension to a full solution The evaluation function f(x^p) can be computed using
function recorded by the Mini-Bucket scheme
B: P(E|B,C) P(D|A,B) P(B|A)
A:
E:
D:
C: P(C|A) hB(E,C)
hB(D,A)
hC(E,A)
P(A) hE(A) hD(A)
f(a,e,D) = P(a) · hB(D,a) · hC(e,a)
g h – is admissible
A
B C
D
E
Belief Network
E
E
DA
D
BD
B
0
1
1
0
1
0
f(a,e,D))=g(a,e) · H(a,e,D )
Heuristics Properties
MB Heuristic is monotone, admissible
Retrieved in linear time IMPORTANT:
Heuristic strength can vary by MB(i). Higher i-bound more pre-processing
stronger heuristic less search.
Allows controlled trade-off between preprocessing and search
Experimental Methodology
Algorithms BBMB(i) – Branch and Bound with MB(i) BBFB(i) - Best-First with MB(i) MBE(i)
Test networks: Random Coding (Bayesian) CPCS (Bayesian) Random (CSP)
Measures of performance Compare accuracy given a fixed amount of time - how
close is the cost found to the optimal solution Compare trade-off performance as a function of time
Empirical Evaluation of mini-bucket heuristics,Bayesian networks, coding
Time [sec]
0 10 20 30
% S
olve
d E
xact
ly
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
BBMB i=2
BFMB i=2
BBMB i=6
BFMB i=6
BBMB i=10
BFMB i=10
BBMB i=14
BFMB i=14
Random Coding, K=100, noise=0.28 Random Coding, K=100, noise 0.32
Time [sec]
0 10 20 30
% S
olve
d E
xact
ly
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
BBMB i=6 BFMB i=6 BBMB i=10 BFMB i=10 BBMB i=14 BFMB i=14
Random Coding, K=100, noise=0.32
38
Max-CSP experiments(Kask and Dechter, 2000)
Dynamic MB Heuristics
Rather than pre-compiling, the mini-bucket heuristics can be generated during search
Dynamic mini-bucket heuristics use the Mini-Bucket algorithm to produce a bound for any node in the search space
(a partial assignment, along the given variable ordering)
Dynamic MB and MBTE Heuristics(Kask, Marinescu and Dechter, 2003)
Rather than precompile compute the heuristics during search
Dynamic MB: Dynamic mini-bucket heuristics use the Mini-Bucket algorithm to produce a bound for any node during search
Dynamic MBTE: We can compute heuristics simultaneously for all un-instantiated variables using mini-bucket-tree elimination .
MBTE is an approximation scheme defined over cluster-trees. It outputs multiple bounds for each variable and value extension at once.
ABC
2
4
),|()|()(),()2,1( bacpabpapcbha
1
3 BEF
EFG
),(),|()|(),( )2,3(,
)1,2( fbhdcfpbdpcbhfd
),(),|()|(),( )2,1(,
)3,2( cbhdcfpbdpfbhdc
),(),|(),( )3,4()2,3( fehfbepfbhe
),(),|(),( )3,2()4,3( fbhfbepfehb
),|(),()3,4( fegGpfeh e
EF
BF
BC
BCDF
G
E
F
C D
B
A
Cluster Tree Elimination - example
Mini-Clustering
Motivation: Time and space complexity of Cluster Tree Elimination
depend on the induced width w* of the problem When the induced width w* is big, CTE algorithm becomes
infeasible The basic idea:
Try to reduce the size of the cluster (the exponent); partition each cluster into mini-clusters with less variables
Accuracy parameter i = maximum number of variables in a mini-cluster
The idea was explored for variable elimination (Mini-Bucket)
Split a cluster into mini-clusters => bound complexity
)()( :decrease complexity lExponentia rnrn eOeO)O(e
Idea of Mini-Clustering
},...,,,...,{ 11 nrr hhhh )(ucluster
elim
n
iihh
1
},...,{ 1 rhh },...,{ 1 nr hh
elim
n
rii
elim
r
ii hhg
11
gh
EF
BF
BC
),|()|()(:),(1)2,1( bacpabpapcbh
a
)2,1(H
fd
fd
dcfpch
fbhbdpbh
,
2)1,2(
,
1)2,3(
1)1,2(
),|(:)(
),()|(:)(
)1,2(H
dc
dc
dcfpfh
cbhbdpbh
,
2)3,2(
,
1)2,1(
1)3,2(
),|(:)(
),()|(:)(
)3,2(H
),(),|(:),( 1)3,4(
1)2,3( fehfbepfbh
e
)2,3(H
)()(),|(:),( 2)3,2(
1)3,2(
1)4,3( fhbhfbepfeh
b
)4,3(H
),|(:),(1)3,4( fegGpfeh e)3,4(H
ABC
2
4
1
3 BEF
EFG
BCDF
Mini-Clustering - example
ABC
2
4
),()2,1( cbh1
3 BEF
EFG
),()1,2( cbh
),()3,2( fbh
),()2,3( fbh
),()4,3( feh
),()3,4( fehEF
BF
BC
BCDF
),(1)2,1( cbh
)(
)(2
)1,2(
1)1,2(
ch
bh
)(
)(2
)3,2(
1)3,2(
fh
bh
),(1)2,3( fbh
),(1)4,3( feh
),(1)3,4( feh
)2,1(H
)1,2(H
)3,2(H
)2,3(H
)4,3(H
)3,4(H
ABC
2
4
1
3 BEF
EFG
EF
BF
BC
BCDF
Mini Bucket Tree Elimination
Mini-Clustering
Correctness and completeness: Algorithm MC(i) computes a bound (or an approximation) for each variable and each of its values.
MBTE: when the clusters are buckets in BTE.
Branch and Bound w/ Mini-Buckets
BB with static Mini-Bucket Heuristics (s-BBMB) Heuristic information is pre-compiled before search.
Static variable ordering, prunes current variable
BB with dynamic Mini-Bucket Heuristics (d-BBMB)
Heuristic information is assembled during search. Static variable ordering, prunes current variable
BB with dynamic Mini-Bucket-Tree Heuristics (BBBT)
Heuristic information is assembled during search. Dynamic variable ordering, prunes all future variables
Empirical Evaluation Algorithms:
Complete BBBT BBMB
Incomplete DLM GLS SLS IJGP IBP (coding)
Measures: Time Accuracy (% exact) #Backtracks Bit Error Rate (coding)
Benchmarks: Coding networks Bayesian Network
Repository Grid networks (N-by-N) Random noisy-OR networks Random networks
Real World Benchmarks
Average Accuracy and Time. 30 samples, 10 observations, 30 seconds
Empirical Results: Max-CSP Random Binary Problems: <N, K, C, T>
N: number of variables K: domain size C: number of constraints T: Tightness
Task: Max-CSP
BBBT(i) vs. BBMB(i).
BBBT(i) vs BBMB(i), N=100
i=2 i=3 i=4 i=5 i=6 i=7 i=2
Searching the Graph; caching goods
C context(C) = [ABC]
D context(D) = [ABD]
F context(F) = [F]
E context(E) = [AE]
B context(B) = [AB]
A context(A) = [A]
A
E
C
B
F
D
A B f1
0 0 20 1 01 0 11 1 4
A C f2
0 0 30 1 01 0 01 1 1
A E f3
0 0 00 1 31 0 21 1 0
A F f4
0 0 20 1 01 0 01 1 2
B C f5
0 0 00 1 11 0 21 1 4
B D f6
0 0 40 1 21 0 11 1 0
B E f7
0 0 30 1 21 0 11 1 0
C D f8
0 0 10 1 41 0 01 1 0
E F f9
0 0 10 1 01 0 01 1 2
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
0 1
0 1 0 1
0 1 0 1
0 1
5
0 0
2 0 0 4
3 1 5 4 0 2 2 5
30
22 1
2
04
35
3 5 1 3 13
56 4
2 24 1
0 56 4
2 24
0
52
5 2 3 0 30
Searching the Graph; caching goods
C context(C) = [ABC]
D context(D) = [ABD]
F context(F) = [F]
E context(E) = [AE]
B context(B) = [AB]
A context(A) = [A]
A
E
C
B
F
D
A B f1
0 0 20 1 01 0 11 1 4
A C f2
0 0 30 1 01 0 01 1 1
A E f3
0 0 00 1 31 0 21 1 0
A F f4
0 0 20 1 01 0 01 1 2
B C f5
0 0 00 1 11 0 21 1 4
B D f6
0 0 40 1 21 0 11 1 0
B E f7
0 0 30 1 21 0 11 1 0
C D f8
0 0 10 1 41 0 01 1 0
E F f9
0 0 10 1 01 0 01 1 2
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1
0 1
0 1 0 1
0 1 0 1
0 1
0 2 1 0
3 3 1 1 2 2 0 0
8 5 3 1 7 4 2 0
6 5 7 4
5 7
5
0 0
2 0 0 4
3 1 5 4 0 2 2 5
30
22 1
2
04
35
3 5 1 3 13
56 4
2 24 1
0 56 4
2 24 1
0
52
5 2 3 0 30
Outline Introduction
Optimization tasks for graphical models Solving by inference and search
Inference Bucket elimination, dynamic programming Mini-bucket elimination, belief propagation
Search Branch and bound and best-first Lower-bounding heuristics AND/OR search spaces
Hybrids of search and inference Cutset decomposition Super-bucket scheme
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