© j. christopher beck 20081 lecture 15: cp search

© J. Christopher Beck 2008 1

Lecture 15: CP Search


Outline Quick CP Review Standard CP Search Scheduling Specific Branching

Heuristics


Readings

P Ch 5.5, D.2, D.3


Generic CP Algorithm

AssertCommitment

Propagators

Start

SuccessSolution?

MakeHeuristicDecision

BacktrackTechnique

Failure

Nothing toretract?

Dead-end?


Constraint Satisfaction Problem (CSP)

Given: 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

From Lecture 13


Idea #1: Partitioning

Add constraint to the original problem to form a partition: P1, P2, P3, … Partitions are easier to solve Partitions, sub-partitions, sub-sub-

partitions Solution is the best one from all

the partitions

From Lecture 11


Standard CP Search

Tree Search a, b, c є {0, 1}

a = 0 a = 1

b = 0 b = 1

c = 0 c = 1 c = 0 c = 1 c = 0 c = 1c = 0 c = 1

b = 0 b = 1

Branch

In CSP there is no optimization function!(think of the Crystal Maze)


Standard CP Search

Branch-and-Infer at each node, run constraint propagation

For Constraint Optimization Problems: Branch-Infer-&-Bound

Partition/Inference/Relax (see Lecture 11) Often bound calculation is “hidden” in

constraint propagation


Basic MIP B&B

Q.push(root)

while Q is not empty

n = Q.pop()

solve LP(n)

if better than incumbent

if integral

update incumbent

else

v = choose branching variable

Q.push(children(n,v))

Priority queue(node selection)

Relaxation

Branching variableselection

From Lecture 12


Basic CP BI&B

CPSearch(Problem P)

if propagate(P) == dead-end

return dead-end

if not all variables are assigned

V = choose variable in P

x = choose value for V

if CPSearch(P + V=x) == solution

return solution

else return CPSearch(P + V≠x)

return solution

depth-firstsearch

variable orderingheuristic

value orderingheuristic


MIP vs CP Search

Conceptually (almost) the same thing

Nice exam question: How do node selection & branch

selection correspond to DFS, variable ordering, and value ordering?


CP Heuristics for Scheduling

It is common to make a decision (branch) about the sequence of a pair of activities. What pair? Which sequence should be tried first?

How do these map into variable and value ordering heuristics?


Slack

slack(Ai,Aj) = lftj – esti – pi – pj

A2 20

35 100

A1 15

12050

slack(A1,A2) = 100 – 50 – 15 – 20 = 15

slack(A2,A1) = 120 – 35 – 15 – 20 = 50

If you post AiAj, howmuch time is left between

the end of Aj and lftj?


MinSlack Heuristic

Find operation pair with smallest slack Post the opposite sequence Example

slack(A B) = 50, slack(B A) = 25 slack(A C) = 150, slack(C A) = 5 slack(B C) = 15, slack(C B) = 50

Pick A,C and post A C

Note: The text is wrong here!


Branch on Sequence

A1 A2 A2 A1

…A4 A1

…


CBA & Slack Find all CBA inferences Find first heuristic decision

based on slack

A3 20

40

50

A1 10

10

50A2 5

A4 5

20

30 50

0


Generic CP Algorithm

AssertCommitment

Propagators

Start

SuccessSolution?

MakeHeuristicDecision

BacktrackTechnique

Failure

Nothing toretract?

Dead-end?


CP on JSP

Run CP on our JSP problem Use

CBA, EF Exclusion Min Slack Heuristic

Jobs Processing times

0 J0R0[15] J0R1[50] J0R2[60]

1 J1R1[50] J1R0[50] J1R2[15]

2 J2R0[30] J2R1[15] J2R2[20]

© j. christopher beck 20081 lecture 15: cp search

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