This document discusses using the Branch and Bound technique to solve the traveling salesman problem and water jug problem. Branch and Bound is a method for solving discrete and combinatorial optimization problems by breaking the problem into smaller subsets, calculating bounds on the objective function, and discarding subsets that cannot produce better solutions than the best found so far. The document provides examples of applying Branch and Bound to find the optimal path between states for the water jug problem and the shortest route between cities for the traveling salesman problem.

1. Solving traveling salesman and water
jug problem using Branch and Bound
Technique
Prepared By
Mehta Ishani

2. Introduction
Branch and Bound
 method for solving optimization problems
 approach developed for solving discrete and
combinatorial optimization problems
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3. Problem domain
discrete optimization problems
(decision variables assume discrete values)
combinatorial optimization problems
(choosing the best combination)
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4. Difficulty
No optimality conditions
To ensure optimality perform comparison
Explicit comparison results in NP Completeness
Implicit comparison results partial enumeratoin
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5. Solution
Branch and Bound technique
 breaking up its feasible set into successively smaller
subsets
 calculating bounds on the objective function value
 discard certain subsets
The method was first proposed by A. H. Land and A.
G. Doig in 1960 for discrete programming
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6. Steps
1)Place the start node of zero path length on the queue.
2)Until the queue is empty or a goal node has been found do
the following:
(i) Determine if the s first path in the queue contains a good
node.
(ii) If the first path contains a goal node exit with success.
(iii) If the first path does not contain a good node remove the
path from the queue and form new paths by extending the
removed paths by on step.
(iv)Compute the cost of the new paths and add them to the
queue.
(v) Sort the paths on the queue with lowest-cost path in front.
3)otherwise exit with failure. 8/23/20146

7. Water Jug
Initial state (0,0)
Goal state (2,0)
Path p1:
(0,0) -> (4,0) -> (1,3) -> (1,0) -> (0,1) -> (4,1) -> (2,3) -
> (2,0)
P1
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8. Water Jug
 Path 2:
(0,0) -> (0,3) -> (3,3)->(3,0)->(4,0)->(0,0)
Since this results infinite loop then not to put in queue
 Path 3:
(0,0) -> (0,3) -> (3,3) -> (4,2) -> (0,2) -> (2,0)
P1
P3 P1
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9. Water Jug
 Now path 1 contains 8 states and path 2 contains 6
states after performing sorting the queue is
 Thus our optimal solution is Path 3
P1 P3
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10. Traveling Salesman Problem
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11. Problem
 NP-hard
 Brute Force takes O(n!) to complete
 Algorithms to solve take one of two forms:
 Exact algorithms
 Branch and Bound O(2n)
 Linear Programming
 Heuristic algorithms
 Nearest neighbor O( log n )
 Pair wise exchange
 Randomized improvement (genetic algorithms)
 Many real world applications:
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12. Solution
 Special properties of the Traveling Salesman
Problem that make it suitable for Branch and Bound:
 As you build your solution, cost increases.
 Partial solutions have valid costs (lower bound costs)
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13. Steps
 Given a complete, weighted graph on n nodes, find the
least weight Hamiltonian cycle, a cycle that visits every
node once.
 Though this problem is easy enough to explain, it is very
difficult to solve.
 Finding all the Hamiltonian cycles of a graph takes
exponential time. Therefore, TSP is in the class NP.
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14. Solution
 Path1 {A, B, C, D, E, A} - length 24
P1
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15. Solution
Path2 {A, B, C, E, D, A} - length 31
P2 P1
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16. Solution
Path 3 {A,B,D,C,E,A} - length 21
P3 P2 P1
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17. Solution
 After perform sorting queue is
 Hence Optimal Solution is Path 3
P2 P1 P3
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18. Pros and Cons
 An enhancement of backtracking
 However, it is much slower. Indeed, it often leads to
exponential time complexities in the worst case.
 On the other hand, if applied carefully, it can lead to
algorithms that run reasonably fast on average.
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19. Summery
 Thus Branch and Bound is:
 a general search method.
 minimize a function f(x), where x is restricted to some
feasible region.
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