Lecture 15: CP Search
© J. Christopher Beck 2008
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Outline
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Quick CP Review
Standard CP Search
Scheduling Specific Branching Heuristics
© J. Christopher Beck 2008
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Readings
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P Ch 5.5, D.2, D.3
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Generic CP Algorithm
Start
Solution?
Propagators
Success
Dead-end?
Backtrack
Technique
Nothing to
retract?
Failure
© J. Christopher Beck 2008
Make
Heuristic
Decision
Assert
Commitment
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Constraint Satisfaction
Problem (CSP)
From Lecture 13
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Given:
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V, a set of variables {v0, v1, …, vn}
D, a set of domains {D0, D1, …, Dn}
C, a set of constraints {c0, c1, …, cm}
Each constraint, ci, has a scope
ci(v0, v2, v4, v117, …), the variables that
it constrains
© J. Christopher Beck 2008
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From Lecture 11
Idea #1: Partitioning
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Add constraint to the original problem
to form a partition: P1, P2, P3, …
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Partitions are easier to solve
Partitions, sub-partitions, sub-subpartitions
Solution is the best one from all the
partitions
© J. Christopher Beck 2008
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Standard CP Search
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Tree Search
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a, b, c є {0, 1}
a=0
b=0
c=0
c=1
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Branch
a=1
b=1
c=0
b=0
c=1
c=0
c=1
b=1
c=0
c=1
In CSP there is no optimization function!
(think of the Crystal Maze)
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Standard CP Search
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Branch-and-Infer
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at each node, run constraint propagation
For Constraint Optimization Problems:
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Branch-Infer-&-Bound
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Partition/Inference/Relax (see Lecture 11)
Often bound calculation is “hidden” in
constraint propagation
© J. Christopher Beck 2008
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From Lecture 12
Basic MIP B&B
Priority queue
(node selection)
Q.push(root)
while Q is not empty
Relaxation
n = Q.pop()
solve LP(n)
if better than incumbent
if integral
Branching variable
update incumbent
selection
else
v = choose branching variable
Q.push(children(n,v))
© J. Christopher Beck 2008
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Basic CP BI&B
CPSearch(Problem P)
if propagate(P) == dead-end
return dead-end
if not all variables are assigned
value ordering V = choose variable in P variable ordering
heuristic
heuristic
x = choose value for V
if CPSearch(P + V=x) == solution
depth-first
return solution
search
else return CPSearch(P + V≠x)
return solution
© J. Christopher Beck 2008
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MIP vs CP Search
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Conceptually (almost) the same thing
Nice exam question:
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How do node selection & branch selection
correspond to DFS, variable ordering, and
value ordering?
© J. Christopher Beck 2008
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CP Heuristics for Scheduling
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It is common to make a decision
(branch) about the sequence of a pair
of activities.
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What pair?
Which sequence should be tried first?
How do these map into variable
and value ordering heuristics?
© J. Christopher Beck 2008
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If you post AiAj, how
much time is left between
the end of Aj and lftj?
Slack
slack(Ai,Aj) = lftj – esti – pi – pj
A1 15
A2 20
35
50
100
120
slack(A1,A2) = 100 – 50 – 15 – 20 = 15
slack(A2,A1) = 120 – 35 – 15 – 20 = 50
© J. Christopher Beck 2008
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Note: The text is wrong here!
MinSlack Heuristic
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Find operation pair with smallest slack
Post the opposite sequence
Example
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slack(A  B) = 50, slack(B  A) = 25
slack(A  C) = 150, slack(C  A) = 5
slack(B  C) = 15, slack(C  B) = 50
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Pick A,C and post A  C
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Branch on Sequence
A1  A2
A4  A1
A2  A1
…
…
© J. Christopher Beck 2008
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CBA & Slack
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Find all CBA inferences
Find first heuristic decision
based on slack
A1 10
20
50
A2 5
30
A3 20
50
0
A4 5
10
© J. Christopher Beck 2008
50
40
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Generic CP Algorithm
Start
Solution?
Propagators
Success
Dead-end?
Backtrack
Technique
Nothing to
retract?
Failure
© J. Christopher Beck 2008
Make
Heuristic
Decision
Assert
Commitment
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CP on JSP
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Run CP on our JSP problem
Use
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CBA, EF Exclusion
Min Slack Heuristic
Jobs
Processing times
0
J0R0[15]  J0R1[50]  J0R2[60]
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J1R1[50]  J1R0[50]  J1R2[15]
2
J2R0[30]  J2R1[15]  J2R2[20]
© J. Christopher Beck 2008
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