Branch and bound

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 BFS-like
search.
• Dead-node: A node that has been expanded
• Solution-node
 LC-Search (Least Cost Search):
• The selection rule for the next E-node in FIFO or LIFO
branch-and-bound is sometimes “blind”. i.e. the selection
1
2 3 4 5
6 7 8 9
1
2 3 4 5
6 78 9
1
2 3 4 5
Live Node: 2, 3, 4, and 5
FIFO Branch & Bound (BFS)
Children of E-node are inserted
in a queue.
LIFO Branch & Bound (D-Search)
Children of E-node are inserted in a
stack.

rule does not give any preference to a node that has a very
good chance of getting the search to an answer node quickly.
• The search for an answer node can often be speeded by using
an “intelligent” ranking function, also called an
approximate cost function
C
^
• Expanded-node (E-node): is the live node with the best
C
^
value
 Requirements
• Branching: A set of solutions, which is represented by a
node, can be partitioned into mutually exclusive sets.
Each subset in the partition is represented by a child of the
original node.
• Lower bounding: An algorithm is available for calculating
a lower bound on the cost of any solution in a given
subset.
 Searching: Least-cost search (LC)
• Cost and approximation
 Each node, X, in the search tree is associated with a
cost: C(X)

 C(X) = cost of reaching the current node, X (E-
node), from the root + the cost of reaching an
answer node from X.
C(X) = g(X) + h(X)
 Get an approximation of C(x),
C
^
(x) such that
C
^
(x) ≤C(x), and
C
^
(x) = C(x) if x is a solution-node.
 The approximation part of
C
^
(x) is
h(x)=the cost of reaching a solution-node from X,
not known.
• Least-cost search:
The next E-node is the one with least
C
^
 Example: 8-puzzle
• Cost function:
C
^
= g(x) +h(x)
where
h(x) = the number of misplaced tiles
and g(x) = the number of moves so far

• Assumption: move one tile in any direction cost 1.
12356784 12358674
Initial State Final State
532
^
C =+=
12358674
12356784
12356478 12356784 12563784
12358674 12356784 13526784
12358674
312
^
C =+=
541
^
C =+=
321
^
C =+= 541
^
C =+=
532
^
C =+=
303
^
C =+=
523
^
C =+=

Note: In case of tie, choose the leftmost node.

 Algorithm:
/* live_node_set: set to hold the live nodes at any time */
/* lowcost: variable to hold the cost of the best cost at any
given node */
Begin
Lowcost = ∞;
While live_node_set ≠∞ do
- choose a branching node, k, such that
k ∈live_node_set; /* k is a E-node */
- live_node_set= live_node_set - {k};
- Generate the children of node k and the
corresponding lower bounds;
Sk={(i,zi): i is child of k and zi its lower
bound}
- For each element (i,zi) in Sk do
- If zi U
- then
- Kill child i; /* i is a child node */
- Else
If child i is a solution
Then
U =zi; current best = child i;
Else
Add child i to live_node_set;
Endif;
Endif;
- Endfor;
Endwhile;

 Travelling Salesman Problem: A Branch and Bound algorithm
• Definition: Find a tour of minimum cost starting from a
node S going through other nodes only once and returning
to the starting point S.
• Definitions:
 A row(column) is said to be reduced iff it contains at
least one zero and all remaining entries are non-
negative.
 A matrix is reduced iff every row and column is
reduced.
• Branching:
 Each node splits the remaining solutions into two
groups: those that include a particular edge and
those that exclude that edge
 Each node has a lower bound.
 Example: Given a graph G=(V,E), let i,j ∈ E,
All Solutions
Solutions with i,j Solutions without i,j
L1
L
L2

• Bounding: How to compute the cost of each node?
 Subtract of a constant from any row and any column
does not change the optimal solution (The path).
 The cost of the path changes but not the path itself.
 Let A be the cost matrix of a G=(V,E).
 The cost of each node in the search tree is computed
as follows:
• Let R be a node in the tree and A(R) its
reduced matrix
• The cost of the child (R), S:
• Set row i and column j to infinity
• Set A(j,1) to infinity
• Reduced S and let RCL be the
reduced cost.
• C(S) = C(R) + RCL+A(i,j)
 Get the reduced matrix A' of A and let L be the
value subtracted from A.
 L: represents the lower bound of the path solution
 The cost of the path is exactly reduced by L.
• What to determine the branching edge?
 The rule favors a solution through left subtree
rather than right subtree, i.e., the matrix is reduced
by a dimension.

 Note that the right subtree only sets the branching
edge to infinity.
 Pick the edge that causes the greatest increase in
the lower bound of the right subtree, i.e., the
lower bound of the root of the right subtree is
greater.
• Example:
o The reduced cost matrix is done as follows:
- Change all entries of row i and column j to
infinity
- Set A(j,1) to infinity (assuming the start node is 1)
- Reduce all rows first and then column of the
resulting matrix

• Given the following cost matrix:
• State Space Tree:
Vertex = 3 Vertex = 5
6 7 8
10
4 535 53 25
Vertex = 3
3
Vertex = 4Vertex = 3
28 50 36
52 28
251
2 31
9
11 28
Vertex = 3

• The TSP starts from node 1: Node 1
o Reduced Matrix: To get the lower bound of the
path starting at node 1
 Row # 1: reduce by 10
 Row #2: reduce 2
 Row #3: reduce by 2

 Row # 4: Reduce by 3:
 Row # 4: Reduce by 4
 Column 1: Reduce by 1
 Column 2: It is reduced.
 Column 3: Reduce by 3

 The reduced cost is: RCL = 25
 So the cost of node 1 is:
• Cost(1) = 25
 The reduced matrix is:
cost(1) = 25

• Choose to go to vertex 2: Node 2
- Cost of edge 1,2 is: A(1,2) = 10
- Set row #1 = inf since we are choosing edge 1,2
- Set column # 2 = inf since we are choosing edge
1,2
- Set A(2,1) = inf
- The resulting cost matrix is:
- The matrix is reduced:
o RCL = 0
- The cost of node 2 (Considering vertex 2 from
vertex 1) is:
 Cost(2) = cost(1) + A(1,2) = 25 + 10 = 35

- Cost of edge 1,3 is: A(1,3) = 17 (In the reduced
matrix
- Set row #1 = inf since we are starting from node 1
- Set column # 3 = inf since we are choosing edge
1,3
- Set A(3,1) = inf
- The resulting cost matrix is:
• Reduce the matrix:
o Rows are reduced
o The columns are reduced except for column # 1:
 Reduce column 1 by 11:

• The lower bound is:
o RCL = 11
• The cost of going through node 3 is:
o cost(3) = cost(1) + RCL + A(1,3) = 25 + 11 + 17
= 53

o Remember that the cost matrix is the one that was
reduced at the starting vertex 1
o Cost of edge 1,4 is: A(1,4) = 0
o Set row #1 = inf since we are starting from node
1
o Set column # 4 = inf since we are choosing edge
1,4
o Set A(4,1) = inf
o The resulting cost matrix is:
o Reduce the matrix:
 Rows are reduced
 Columns are reduced
o The lower bound is: RCL = 0
o The cost of going through node 4 is:

 cost(4) = cost(1) + RCL + A(1,4) = 25 + 0
+ 0 = 25

o Remember that the cost matrix is the one that was
reduced at starting vertex 1
o Set row #1 = inf since we are starting from node
1
o Set column # 5 = inf since we are choosing edge
1,5
o Set A(5,1) = inf
 Reduce rows:
• Reduce row #2: Reduce by 2

• Reduce row #4: Reduce by 3
o The lower bound is:
 RCL = 2 + 3 = 5
 cost(5) = cost(1) + RCL + A(1,5) = 25 + 5 +
1 = 31

• In summary:
o So the live nodes we have so far are:
 2: cost(2) = 35, path: 1-2
 3: cost(3) = 53, path: 1-3
 4: cost(4) = 25, path: 1-4
 5: cost(5) = 31, path: 1-5
o Explore the node with the lowest cost: Node 4
has a cost of 25
o Vertices to be explored from node 4: 2, 3, and 5
o Now we are starting from the cost matrix at node
4 is:
Cost(4) = 25

• Choose to go to vertex 2: Node 6 (path is 1-4-2)
o Set row #4 = inf since we are considering edge
4,2
o Set column # 2 = inf since we are considering
edge 4,2
o Set A(2,1) = inf
 Rows are reduced
 cost(6) = cost(4) + RCL + A(4,2) = 25 + 0 +
3 = 28

• Choose to go to vertex 3: Node 7 ( path is 1-4-3 )
4,3
edge 4,3
o Set A(3,1) = inf
 Reduce row #3: by 2:
 Reduce column # 1: by 11

o So the RCL of node 7 (Considering vertex 3 from
vertex 4) is:
 Cost(7) = cost(4) + RCL + A(4,3) = 25 + 13
+ 12 = 50
• Choose to go to vertex 5: Node 8 ( path is 1-4-5 )
4,5
edge 4,5
o Set A(5,1) = inf

 Reduced row 2: by 11
o So the cost of node 8 (Considering vertex 5 from
vertex 4) is:
 Cost(8) = cost(4) + RCL + A(4,5) = 25 + 11
+ 0 = 36

• In summary:
 2: cost(2) = 35, path: 1-2
 3: cost(3) = 53, path: 1-3
 5: cost(5) = 31, path: 1-5
 6: cost(6) = 28, path: 1-4-2
 7: cost(7) = 50, path: 1-4-3
 8: cost(8) = 36, path: 1-4-5
has a cost of 28
o Vertices to be explored from node 6: 3 and 5
6 is:
Cost(6) = 28

• Choose to go to vertex 3: Node 9 ( path is 1-4-2-
3 )
2,3
edge 2,3
o Set A(3,1) = inf
 Reduce row #3: by 2

 Reduce column # 1: by 11
o The lower bound is: RCL = 2 +11 = 13
o So the cost of node 9 (Considering vertex 3 from
vertex 2) is:
 Cost(9) = cost(6) + RCL + A(2,3) = 28 + 13
+ 11 = 52
5 )

2,3
edge 2,3
o Set A(5,1) = inf
 Rows reduced
 Columns reduced
o So the cost of node 10 (Considering vertex 5
from vertex 2) is:
 Cost(10) = cost(6) + RCL + A(2,3) = 28 + 0
+ 0 = 28

• In summary:
 2: cost(2) = 35, path: 1-2
 3: cost(3) = 53, path: 1-3
 5: cost(5) = 31, path: 1-5
 7: cost(7) = 50, path: 1-4-3
 8: cost(8) = 36, path: 1-4-5
 9: cost(9) = 52, path: 1-4-2-3
 10: cost(2) = 28, path: 1-4-2-5
has a cost of 28
o Vertices to be explored from node 10: 3
10 is:

5-3 )
5,3
edge 5,3
o Set A(3,1) = inf
 Rows reduced
 Columns reduced
o So the cost of node 11 (Considering vertex 5
from vertex 3) is:

 Cost(11) = cost(10) + RCL + A(5,3) = 28 +
0 + 0 = 28
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Branch and bound

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