The document compares and contrasts the Backtracking and Branch & Bound algorithms. Backtracking searches the entire state space tree using depth-first search until it finds a solution, while realizing when it has made an incorrect choice. Branch & Bound may search the tree using depth-first or breadth-first search, and prunes branches when it finds a better solution than exploring that branch could provide. The document also provides an example of applying the Branch & Bound algorithm to the Traveling Salesman Problem and explains how to compute the costs of nodes in this problem.

1. Traveling salesman

2. What are the differences between 'Backtracking'
and 'Branch & Bound' Algorithm techniques?
Backtracking
[1] It is used to find all possible
solutions available to the problem.
[2] It traverse tree by DFS (Depth First
Search).
[3] It realizes that it has made a bad
choice & undoes the last choice by
backing up.
[4] It search the state space tree until it
found a solution.
[5] It involves feasibility function
Branch-and-Bound
[1] It is used to solve optimization problem.
[2] It may traverse the tree in any manner,
DFS or BFS.
[3] It realizes that it already has a better
optimal solution that the pre-solution leads to
so it abandons that pre-solution.
[4] It completely searches the state space tree
to get optimal solution.
[5] It involves bounding function.

3. Here is the state space tree for n=4 (Using Branch And Bound)

4. What is traveling salesman problem?
• Let us look at a situation that there are 5 cities,
which are represented as NODES.
• There is a Person at NODE-1
• This person has to reach each nodes one and only
ones and come back to original (starting) position
• This process has to occur with minimum cost or
minimum distance travelled.
• Note that starting point can start with any Node. For
Example:
1-2-4-5-3-1
1-5-2-3-4-1

5. How to compute the cost of each node?
• A row/column is said to be reduce 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.
Rules of reduced cost matrix :
1. Change all entries in row i and column j from A to inf
2. Set A[j,1] to inf
3. Reduce all rows and columns in the resulting except for rows and
column containing only inf.
4. Cost Function C(s)=C(R)+A[i,j]+ r [r – total reduce value]

6. -10
-2
-2
-3
-4
REDUCTION (ROW)
INF 20 30 10 11
15 INF 16 4 2
3 5 INF 2 4
19 6 18 INF 3
16 4 7 16 INF
-1 -3
COLUMN REDUCTION
TOTAL REDUCTION = (10+2+2+3+4)+(1+3)
= 25
INF 10 17 0 1
12 INF 11 2 0
1 3 INF 0 2
15 3 12 INF 0
11 0 3 12 INF
INF 10 17 0 1
12 INF 11 2 0
0 3 INF 0 2
15 3 12 INF 0
11 0 0 12 INF

10. Thank you all…….