Branch_and_Bound.doc

8/12/2019 Branch_and_Bound.doc

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Branch and Bound

Definitions:

Branch and Bound is a state space search method in which

all the children of a node are generated before expanding

any of its children.

Live-node: A node that has not been expanded.

It is similar to backtracking technique but uses BF!like

search.

Dead!node: A node that has been expanded

olution!node

"#!earch $"east #ost earch%:

&he selection rule for the next '!node in FIF( or "IF(

branch!and!bound is sometimes )blind*. i.e. the selection

+

, - /

0 1 2 3

+

, - /

0 12 3

+

, - /

"i4e 5ode: ,6 -6 6 and /

FIF( Branch 7 Bound $BF%

#hildren of '!node are inserted

in a queue.

"IF( Branch 7 Bound $D!earch%

#hildren of '!node are inserted in a

stack.


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rule does not gi4e any preference to a node that has a 4ery

good chance of getting the search to an answer node quickly.

&he search for an answer node can often be speeded by using

an )intelligent* ranking function6 also called an

approximate cost function#

8

'xpanded!node $'!node%: is the li4e node with the best#

8

4alue

9equirements

Branching: A set of solutions6 which is represented by a

node6 can be partitioned into mutually exclusi4e sets.

'ach subset in the partition is represented by a child of the

original node.

"ower bounding: An algorithm is a4ailable for calculating

a lower bound on the cost of any solution in a gi4en

subset.

earching: "east!cost search $"#%

#ost and approximation

'ach node6 6 in the search tree is associated with a

cost: #$%


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#$% ; cost of reaching the current node6 $'!

node%6 from the root+ the cost of reaching an

answer node from .

#$% ; g$% < h$%

=et an approximation of #$x%6#

8$x% such that

#

8$x% #$x%6 and

#

8$x% ; #$x% if x is a solution!node.

&he approximation part of#

8$x% is

h$x%;the cost of reaching a solution!node from 6

not known.

"east!cost search:

&he next '!node is the one with least#

8

'xample: 2!pu>>le

#ost function: #

8

; g$x%


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Assumption: mo4e one tile in any direction cost +.

12356784 12358674

Initial State inal State

/-,

8

# =+=

12358674

12356784

1235647812356784

12563784

12358674 12356784 13526784

12358674

-+,

8

# =+=

/.+

8

# =+=-,+

8

# =+= /.+

8

# =+=

/-,

8

# =+=

-?-

8

# =+=

/,-

8

# =+=


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5ote: In case of tie6 choose the leftmost node.


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Algorithm:

@ li4enodeset: set to hold the li4e nodes at any time @

@ lowcost: 4ariable to hold the cost of the best cost at any

gi4en node @

Begin"owcost ; C

hile li4enodeset do

! choose a branching node6 k6 such that

k li4enodesetC @ k is a '!node @

- li4enodeset; li4enodeset ! EkC

- =enerate the children of node k and the

corresponding lower boundsC

k;E$i6>i%: i is child of k and >iits lower

bound

- For each element $i6>i% in kdo

- If >iG H

- then

- ill child iC @ i is a child node @

- 'lse

If child i is a solution&hen

H ;>iC current best ; child iC

'lse

Add child i to li4enodesetC

'ndifC

'ndifC

! 'ndforC

'ndwhileC


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&ra4elling alesman Jroblem: ! "ranc# and "ound al$orit#m

Definition: Find a tour of minimum cost starting from a

node going through other nodes only once and returning

to the starting point .

Definitions:

A row$column% is said to be reduced iff it contains at

least one >ero and all remaining entries are non!

negati4e.

A matrix is reduced iff e4ery row and column isreduced.

"ranc#in$:

'ach node splits the remaining solutions into two

groups: those that include a particular edge and

those that exclude that edge

'ach node has a lower bound.

'xample: =i4en a graph =;$K6'%6 let Li6MG '6

!ll Solutions

Solutions %it# &i'() Solutions %it#out &i'()

"+

"

",


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"oundin$: Now to compute the cost of each nodeO

ubtract of a constant from any row and any column

does not change the optimal solution $&he path%.

&he cost of the path changes but not the path itself.

"et A be the cost matrix of a =;$K6'%.

&he cost of each node in the search tree is computed

as follows:

"et 9 be a node in the tree and A$9% its

reduced matrix

&he cost of the child $9%6 :

et row i and column M to infinity

et A$M6+% to infinity

9educed and let 9#" be the

reduced cost.

#$% ; #$9% < 9#"


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=i4en the following cost matrix:

tate pace &ree:

Kertex ; - Kertex ; /

0 1 2

+?

/-/ /- ,/

Kertex ; , Kertex ; /Kertex ; -

-

Kertex ; , Kertex ; /Kertex ; Kertex ; -

,2 /? -0

/, ,2

,/+

,

3

++ ,2

Kertex ; -


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&he &J starts from node +: *ode 1

o 9educed Qatrix: &o get the lower bound of the

path starting at node +

9ow R +: reduce by +?

9ow R,: reduce ,

9ow R-: reduce by ,


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9ow R : 9educe by -:

9ow R : 9educe by

#olumn +: 9educe by +

#olumn ,: It is reduced.

#olumn -: 9educe by -


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#olumn : It is reduced.

#olumn /: It is reduced.

&he reduced cost is: 9#" ; ,/

o the cost of node + is:

#ost$+% ; ,/

&he reduced matrix is:

cost1,; ,/


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#hoose to go to 4ertex ,: *ode 2

- #ost of edge L+6,G is: A$+6,% ; +?

- et row R+ ; inf since we are choosing edge L+6,G

- et column R , ; inf since we are choosing edge

L+6,G

- et A$,6+% ; inf

-&he resulting cost matrix is:

- &he matrix is reduced:

o 9#" ; ?

- &he cost of node , $#onsidering 4ertex , from

4ertex +% is:

ost2, . cost1, + !1'2, . 25 + 1/ . 35


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#hoose to go to 4ertex -: *ode 3

- #ost of edge L+6-G is: A$+6-% ; +1 $In the reduced

matrix

- et row R+ ; inf since we are starting from node +

- et column R - ; inf since we are choosing edge

L+6-G

- et A$-6+% ; inf

- &he resulting cost matrix is:

9educe the matrix:

o 9ows are reduced

o &he columns are reduced except for column R +:

9educe column + by ++:


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&he lower bound is:

o 9#" ; ++

&he cost of going through node - is:

o cost$-% ; cost$+% < 9#" < A$+6-% ; ,/ < ++ < +1

; /-


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#hoose to go to 4ertex : *ode 4

o 9emember that the cost matrix is the one that was

reduced at the starting 4ertex +

o #ost of edge L+6G is: A$+6% ; ?

o et row R+ ; inf since we are starting from node

+

o et column R ; inf since we are choosing edge

L+6G

o et A$6+% ; inf

o &he resulting cost matrix is:

o 9educe the matrix:

9ows are reduced

#olumns are reduced

o &he lower bound is: 9#" ; ?

o &he cost of going through node is:


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cost$% ; cost$+% < 9#" < A$+6% ; ,/ < ?

< ? ; ,/


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#hoose to go to 4ertex /: *ode 5

o 9emember that the cost matrix is the one that was

reduced at starting 4ertex +

o #ost of edge L+6/G is: A$+6/% ; +

o et row R+ ; inf since we are starting from node

+

o et column R / ; inf since we are choosing edge

L+6/G

o et A$/6+% ; inf

o &he resulting cost matrix is:

o 9educe the matrix:

9educe rows:

9educe row R,: 9educe by ,


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9educe row R: 9educe by -

#olumns are reduced

o &he lower bound is:

9#" ; , < - ; /

o &he cost of going through node / is:

cost$/% ; cost$+% < 9#" < A$+6/% ; ,/ < /

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