Constraint Optimization

1
Chapter 13

• Constraint Optimization
• And counting, and enumeration
• 275 class

2
From satisfaction to counting and
enumeration and to general graphical models

• ICS-275Spring 2007

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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

4
Constraint Satisfaction

• Example map coloring
• Variables - countries (A,B,C,etc.)
• Values - colors (e.g., red, green, yellow)
• Constraints

Task consistency? Find a solution, all
solutions, counting
5
Propositional Satisfiability
? (C), (A v B v C), (A v B v E), (B v C v
D).
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Constraint Optimization Problemsfor Graphical
Models
f(A,B,D) has scope A,B,D
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Constraint Optimization Problemsfor Graphical
Models
f(A,B,D) has scope A,B,D
Primal graph Variables --gt nodes Functions,
Constraints -? arcs f1(A,B,D) f2(D,F,G) f3(B,C,F
)
F(a,b,c,d,f,g) f1(a,b,d)f2(d,f,g)f3(b,c,f)
G
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Constrained Optimization
Example power plant scheduling
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Probabilistic Networks
P(DC,B)
P(S,C,B,X,D) P(S) P(CS) P(BS) P(XC,S)
P(DC,B)
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Graphical models

• A graphical model (X,D,C)
• X X1,Xn variables
• D D1, Dn domains
• C F1,,Ft functions(constraints, CPTS, cnfs)
• Primal graph

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Graphical models

• A graphical model (X,D,C)
• X X1,Xn variables
• D D1, Dn domains
• C F1,,Ft functions(constraints, CPTS, cnfs)
• Primal graph

A COP problem defined by operators sum, product
and min,max over consistent solutions

• MPE maxX ?j Pj
• CSP ?X ?j Cj
• Max-CSP minX ?j Fj
• Optimization minX ?j Fj

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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

13
Computing MPE
MPE
B
C
E
D
P(a)
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Finding
Algorithm elim-mpe (Dechter 1996)Non-serial
Dynamic Programming (Bertele and Briochi, 1973)
Elimination operator
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Generating the MPE-tuple
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Complexity
Algorithm elim-mpe (Dechter 1996)Non-serial
Dynamic Programming (Bertele and Briochi, 1973)
Elimination operator
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Complexity of bucket elimination
Bucket-elimination is time and space
r number of functions
The effect of the ordering
Finding smallest induced-width is hard
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Directional i-consistency
Adaptive
d-arc
d-path
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Mini-bucket approximation MPE task
Split a bucket into mini-buckets gtbound
complexity
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Mini-Bucket Elimination
P(A)
P(BA)
P(CA)
P(EB,C)
P(DA,B)
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MBE-MPE(i) Algorithm Approx-MPE (DechterRish
1997)

• Input i max number of variables allowed in a
  mini-bucket
• Output lower bound (P of a sub-optimal
  solution), upper bound

Example approx-mpe(3) versus elim-mpe
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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)

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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

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The Search Space
Objective function
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The Search Space
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Arc-cost is calculated based on cost components.
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The Value Function
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Value of node minimal cost solution below it
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An Optimal Solution
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Value of node minimal cost solution below it
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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
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Classic Branch-and-Bound
Upper Bound UB
Lower Bound LB
g(n)
LB(n) g(n) h(n)
n
Prune if LB(n) UB
h(n)
OR Search Tree
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How to Generate Heuristics

• The principle of relaxed models
• Linear optimization for integer programs
• Mini-bucket elimination
• Bounded directional consistency ideas

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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 the probability of its best
extension
D
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)
32
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) lt 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) lt
f(a,e,d) The heuristic function H is what is
compiled during the preprocessing stage of the
Mini-Bucket algorithm.
33
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) gt 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) lt
f(a,e,d) The heuristic function H is what is
compiled during the preprocessing stage of the
Mini-Bucket algorithm.
34
Static MBE Heuristics

• Given a partial assignment xp, estimate the cost
  of the best extension to a full solution
• The evaluation function f(xp) can be computed
  using function recorded by the Mini-Bucket scheme

f(a,e,D))g(a,e) H(a,e,D )
35
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

36
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

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Empirical Evaluation of mini-bucket
heuristics,Bayesian networks, coding
38
Max-CSP experiments(Kask and Dechter, 2000)
39
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)

40
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.

41
Cluster Tree Elimination - example
ABC
1
BC
BCDF
2
BF
BEF
3
EF
EFG
4
42
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)

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Idea of Mini-Clustering
Split a cluster into mini-clusters gt bound
complexity
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Mini-Clustering - example
ABC
1
BC
BCDF
2
BF
BEF
3
EF
EFG
4
45
Mini Bucket Tree Elimination
ABC
ABC
1
1
BC
BC
BCDF
BCDF
2
2
BF
BF
BEF
BEF
3
3
EF
EF
EFG
EFG
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46
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.

47
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

48
Empirical Evaluation

• Measures
• Time
• Accuracy ( exact)
• Backtracks
• Bit Error Rate (coding)

• Algorithms
• Complete
• BBBT
• BBMB
• Incomplete
• DLM
• GLS
• SLS
• IJGP
• IBP (coding)

• Benchmarks
• Coding networks
• Bayesian Network Repository
• Grid networks (N-by-N)
• Random noisy-OR networks
• Random networks

49
Random Bayesian Networks Average Accuracy
Average Accuracy. Random Bayesian (N100, C90,
P2), w 17 100 samples, 10 observations (a) K
2, (b) K 3.
50
Random Bayesian Networks Solution Quality
Solution Quality. Random Bayesian (N100, C90,
P3), w30, 100 samples, 10 observations (a) K
2, (b) K 5.
51
Grid Networks
Average Accuracy. Grid Networks (N100), w
15 100 samples, 10 observations
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Random Coding Networks Bit Error Rate
Average BER. Random Coding (N200, P4),
w22, 100 samples, 60 seconds
53
Real World Benchmarks
Average Accuracy and Time. 30 samples, 10
observations, 30 seconds
54
Empirical Results Max-CSP

• Random Binary Problems ltN, K, C, Tgt
• N number of variables
• K domain size
• C number of constraints
• T Tightness
• Task Max-CSP

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BBBT(i) vs BBMB(i), N50
BBBT(i) vs. BBMB(i)
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BBBT(i) vs BBMB(i), N100
BBBT(i) vs. BBMB(i).
57
Searching the Graph caching goods
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Searching the Graph caching goods
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D context(D) ABD
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F context(F) F
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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

60
Classic OR Search Space
Ordering A B E C D F
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1
61
AND/OR Search Space
A
F
B
C
D
E
Primal graph
DFS tree
62
AND/OR vs. OR
AND
0
1
AND/OR
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
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
AND/OR size exp(4), OR size exp(6)
OR
63
OR space vs. AND/OR space
Random graphs with 20 nodes, 20 edges and 2
values per node.
64
AND/OR vs. OR
(A1,B1) (B0,C0)
AND/OR
F
65
AND/OR vs. OR
(A1,B1) (B0,C0)
AND/OR
Space linear Time O(exp(m)) O(w log n)
Linear space, Time O(exp(n))
F
66
CSP AND/OR Search Tree
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
C
C
OR
C
C
E
E
E
E
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
D
D
D
D
D
D
D
F
F
F
F
F
F
F
F
AND
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
67
CSP AND/OR Search Tree
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
C
C
OR
C
C
E
E
E
E
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
D
D
D
D
D
D
D
F
F
F
F
F
F
F
F
AND
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
68
CSP AND/OR Tree DFS
OR
A
AND
0
OR
B
AND node Combination operator (product)
AND
3
0
OR
C
3
E
1
OR node Marginalization operator (summation)
AND
2
1
1
0
0
1
0
1
OR
D
D
2
1
1
F
AND
0
1
0
1
0
1
1
1
1
0
0
1
69
AND/OR Tree Search for COP
5
A
0
5
0
B
5
2
6
0
C
3
E
3
3
1
3
5
0
1
5
2
0
2
0
1
D
D
F
F
5
2
0
2
0
1
0
1
0
1
0
1
5
6
4
2
3
0
2
2
AND node Combination operator (summation)
OR node Marginalization operator (minimization)
70
Summary of AND/OR Search Trees

• Based on a backbone pseudo-tree
• A solution is a subtree
• Each node has a value cost of the optimal
  solution to the subproblem (computed recursively
  based on the values of the descendants)
• Solving a task finding the value of the root
  node
• AND/OR search tree and algorithms are
• (Freuder Quinn85, Collin, Dechter
  Katz91, Bayardo Miranker95)
• Space O(n)
• Time O(exp(m)), where m is the depth of the
  pseudo-tree
• Time O(exp(w log n))
• BFS is time and space O(exp(w log n)

71
An AND/OR Graph Caching Goods
A
J
A
F
B
B
C
K
C
E
D
E
D
F
G
J
G
H
OR
A
H
K
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
OR
G
G
J
J
AND
0
1
0
1
0
1
0
1
OR
H
H
H
H
K
K
K
K
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
72
Context-based Caching

• Caching is possible when context is the same
• context parent-separator set in induced
  pseudo-graph current variable parents
  connected to subtree below

context(B) A, B context(c)
A,B,C context(D) D context(F) F
73
Complexity of AND/OR Graph

• Theorem Traversing the AND/OR search graphis
  time and space exponential in the induced
  width/tree-width.
• If applied to the OR graph complexity is time and
  space exponential in the path-width.

74
CSP AND/OR Search Tree
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
C
C
OR
C
C
E
E
E
E
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
D
D
D
D
D
D
D
F
F
F
F
F
F
F
F
AND
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
75
CSP AND/OR Tree DFS
OR
A
AND
0
OR
B
AND
3
0
OR
C
3
E
1
AND
2
1
1
0
0
1
0
1
OR
D
D
2
1
1
F
AND
0
1
0
1
0
1
1
1
1
0
0
1
76
CSP AND/OR Search Graph(Caching Goods)
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
C
C
OR
C
C
E
E
E
E
AND
0
1
0
1
0
1
0
1
OR
D
D
D
D
F
F
F
F
AND
0
1
0
1
77
CSP AND/OR Search Graph(Caching Goods)
Time and Space O(exp(w))
OR
A
AND
0
1
OR
B
B
AND
0
1
0
1
C
C
OR
C
C
E
E
E
E
AND
0
1
0
1
0
1
0
1
OR
D
D
D
D
F
F
F
F
AND
0
1
0
1
78
All Four Search Spaces
Full OR search tree 126 nodes
Context minimal OR search graph 28 nodes
Context minimal AND/OR search graph 18 AND nodes
Full AND/OR search tree 54 AND nodes
79
AND/OR vs. OR DFS algorithms
k domain size m pseudo-tree depth n
number of variables w induced width pw path
width

• AND/OR tree
• Space O(n)
• Time O(n km)
• O(n kw log n)
• (Freuder85 Bayardo95 Darwiche01)
• AND/OR graph
• Space O(n kw)
• Time O(n kw)

• OR tree
• Space O(n)
• Time O(kn)
• OR graph
• Space O(n kpw)
• Time O(n kpw)

80
Searching AND/OR Graphs

• AO(i) searches depth-first, cache i-context
• i the max size of a cache table (i.e. number of
  variables in a context)

i0
iw
Space O(exp w) Time O(exp w)
Space O(n) Time O(exp(w log n))
AO(i) time complexity?
81
End of class

• We did not cover the rest of the slides

82
AND/OR Branch-and-Bound (AOBB)

• Associate each node n with a static heuristic
  estimate h(n) of v(n)
• h(n) is a lower bound on the value v(n)
• For every node n in the search tree
• ub(n) current best solution cost rooted at n
• lb(n) lower bound on the minimal cost at n

83
Lower/Upper Bounds
UB(X) best cost below X (i.e. v(X,0))
X
LB(X) LB(X,1)
v(X,0)
1
0
LB(X,1) l(X,1) v(A) h(C) LB(B)
v(A)
h(C)
A
B
C
LB(B) LB(B,0)
h(B,0)
LB(B,0) h(B,0)
0
Prune below AND node (B,0) if LB(X) UB(X)
84
Shallow/Deep Cutoffs
UB(X)
Prune if LB(X) UB(X)
X
LB(X)
1
0
UB(X)
X
A
B
C
LB(X) h(X,1)
0
1
0
Shallow cutoff
E
D
0
Reminiscent of Minimax shallow/deep cutoffs
Deep cutoff
85
Summary of AOBB

• Traverses the AND/OR search tree in a depth-first
  manner
• Lower bounds computed based on heuristic
  estimates of nodes at the frontier of search, as
  well as the values of nodes already explored
• Prunes the search space as soon as an upper-lower
  bound violation occurs

86
Heuristics for AND/OR

• In the AND/OR search space h(n) can be computed
  using any heuristic. We used
• Static Mini-Bucket heuristics
• Dynamic Mini-Bucket heuristics
• Maintaining FDAC Larrosa Schiex03
• (full directional soft arc-consistency)

87
Empirical Evaluation

• Tasks
• Solving WCSPs
• Finding the MPE in belief networks
• Benchmarks (WCSP)
• Random binary WCSPs
• RLFAP networks (CELAR6)
• Bayesian Networks Repository
• Algorithms
• s-AOMB(i), d-AOMB(i), AOMFDAC
• s-BBMB(i), d-BBMB(i), BBMFDAC
• Static variable ordering (dfs traversal of the
  pseudo-tree)

88
Random Binary WCSPs(Marinescu and Dechter, 2005)
D-AOMB vs D-BBMB
S-AOMB vs S-BBMB
Random networks with n20 (number of variables),
d5 (domain size), c100 (number of constraints),
t70 (tightness). Time limit 180 seconds. AO
search is superior to OR search
89
Random Binary WCSPs (contd.)
dense
sparse
n20 (variables), d5 (domain size), c100
(constraints), t70 (tightness)
n50 (variables), d5 (domain size), c80
(constraints), t70 (tightness)
AOMB for large i is competitive with AOMFDAC
90
Resource Allocation
Radio Link Frequency Assignment Problem (RLFAP)
CELAR6 sub-instances
AOMFDAC is superior to ORMFDAC
91
Bayesian Networks Repository
Time limit 600 seconds
available at http//www.cs.huji.ac.il/labs/compbio
/Repository
Static AO is better with accurate heuristic
(large i)
92
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
• Searching trees
• Searching graphs
• Hybrids of search and inference
• Cutset decomposition
• Super-bucket scheme

93
From Searching Trees to Searching Graphs

• Any two nodes that root identical
  subtrees/subgraphs can be merged
• Minimal AND/OR search graph closure under merge
  of the AND/OR search tree
• Inconsistent sub-trees can be pruned too.
• Some portions can be collapsed or reduced.

94
AND/OR Search Graph
context(A) A context(B) B,A context(C)
C,B context(D) D context(E)
E,A context(F) F
A
B
E
C
D
F
Primal graph
Pseudo-tree
95
AND/OR Search Graph
context(A) A context(B) B,A context(C)
C,B context(D) D context(E)
E,A context(F) F
A
B
E
C
D
F
Primal graph
Pseudo-tree
OR
A
AND
0
1
OR
B
B
AND
1
0
0
1
C
C
OR
C
C
E
E
E
E
AND
0
1
1
0
0
1
0
1
OR
D
D
D
D
F
F
F
F
AND
0
1
0
1
96
Context-based caching
context(A) A context(B) B,A context(C)
C,B context(D) D context(E)
E,A context(F) F
A
B
Primal graph
E
C
D
F
Cache Table (C)
Space O(exp(2))
97
Searching AND/OR Graphs

• AO(j) searches depth-first, cache j-context
• j the max size of a cache table (i.e. number of
  variables in a context)

j0
jw
Space O(exp w) Time O(exp w)
Space O(n) Time O(exp(w log n))
AO(j) time complexity?
98
Graph AND/OR Branch-and-Bound - AOBB(j)
Marinescu Dechter, CP2005

• Like BB on tree except, during search, merge
  nodes based on context (caching) maintain cache
  tables of size O(exp(j)), where j is a bound on
  the size of the context.
• Heuristics
• Static mini-bucket
• Dynamic mini-bucket
• Soft directional arc-consistency

99
Pseudo-trees (I)

• AND/OR graph/tree search algorithms influenced by
  the pseudo-tree quality
• Finding the minimal context/depth pseudo-tree is
  a hard problem
• Heuristics
• Min-fill (min context)
• Hypergraph separation (min depth)

100
Pseudo-trees (II)

• MIN-FILL Kjæaerulff, 1990, Bayardo Miranker,
  1995
• depth first traversal of the induced graph
  constructed along some elimination order
• elimination order prefers variables with smallest
  fill set
• HYPERGRAPH Darwiche, 2001
• constraints are vertices in the hypergraph and
  variables are hyperedges
• Recursive decomposition of the hypergraph while
  minimizing the separator size (i.e. number of
  variables) at each step
• Using state-of-the-art software package hMeTiS

101
Quality of pseudo-trees
Bayesian Networks Repository
SPOT5 Benchmarks
102
Empirical Evaluation

• Tasks
• Solving binary WCSPs
• Finding the MPE in belief networks
• Benchmarks
• Random networks
• Resource allocation (SPOT5)
• Genetic linkage analysis
• Algorithms
• s-AOMB(i,j), d-AOMB(i,j), AOMFDAC(j), VEC
• j is the cache bound
• Static variable ordering

103
Random Networks
Static heuristics
Dynamic Heuristics
Caching helps for static heuristics with small
i-bounds
Random networks with n100 (number of variables),
d3 (domain size), c90 (number of CPTs), p2
(number of parents per CPT). Time limit 180
seconds.
104
Resource Allocation SPOT5
Time limit 1800 seconds
Cache means ji. we picked bestperforming i
The heuristics MB(i) is strong enough to prune so
caching is less relevant
105
6 people, 3 markers
L12m
L12f
L11m
L11f
X12
X11
S15m
S13m
L13m
L13f
L14m
L14f
X14
X13
S15m
S15m
L15m
L15f
L16m
L16f
S16m
S15m
X15
X16
L22m
L22f
L21m
L21f
X21
X22
S25m
S23m
L23m
L23f
L24m
L24f
X23
X24
S25m
S25m
L25m
L25f
L26m
L26f
S26m
S25m
X25
X26
L32m
L32f
L31m
L31f
X32
X31
S35m
S33m
L33m
L33f
L34m
L34f
X33
X34
S35m
S35m
L35m
L35f
L36m
L36f
S36m
S35m
X35
X36
106
Empirical evaluation
averages over 5 runs
107
Pedigree 23
averages over 5 runs
108
Pedigree 30
averages over 5 runs
109
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