Branch and bound technique
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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.
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- 3. Problem domain discrete optimization problems (decision variables assume discrete values) combinatorial optimization problems (choosing the best combination) 8/23/20143
- 4. Difficulty No optimality conditions To ensure optimality perform comparison Explicit comparison results in NP Completeness Implicit comparison results partial enumeratoin 8/23/20144
- 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 8/23/20145
- 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 8/23/20147
- 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 8/23/20148
- 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 8/23/20149
- 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: 8/23/201411
- 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) 8/23/201412
- 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. 8/23/201413
- 14. Solution Path1 {A, B, C, D, E, A} - length 24 P1 8/23/201414
- 15. Solution Path2 {A, B, C, E, D, A} - length 31 P2 P1 8/23/201415
- 16. Solution Path 3 {A,B,D,C,E,A} - length 21 P3 P2 P1 8/23/201416
- 17. Solution After perform sorting queue is Hence Optimal Solution is Path 3 P2 P1 P3 8/23/201417
- 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. 8/23/201418
- 19. Summery Thus Branch and Bound is: a general search method. minimize a function f(x), where x is restricted to some feasible region. 8/23/201419
- 20. 8/23/201420