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  1. Getting Started with Algorithm
    What is an Algorithm?
  2. Characteristics of Algorithm
    1 Topic
  3. Analysis Framework
  4. Performance Analysis
    3 Topics
  5. Mathematical Analysis
    2 Topics
  6. Sorting Algorithm
    Sorting Algorithm
    10 Topics
  7. Searching Algorithm
    6 Topics
  8. Fundamental of Data Structures
  9. Queues
  10. Graphs
  11. Trees
  12. Sets
  13. Dictionaries
  14. Divide and Conquer
    General Method
  15. Binary Search
  16. Recurrence Equation for Divide and Conquer
  17. Finding the Maximum and Minimum
  18. Merge Sort
  19. Quick Sort
  20. Stassen’s Matrix Multiplication
  21. Advantages and Disadvantages of Divide and Conquer
  22. Decrease and Conquer
    Insertion Sort
  23. Topological Sort
  24. Greedy Method
    General Method
  25. Coin Change Problem
  26. Knapsack Problem
  27. Job Sequencing with Deadlines
  28. Minimum Cost Spanning Trees
    2 Topics
  29. Single Source Shortest Paths
    1 Topic
  30. Optimal Tree Problem
    1 Topic
  31. Transform and Conquer Approach
    1 Topic
  32. Dynamic Programming
    General Method with Examples
  33. Multistage Graphs
  34. Transitive Closure
    1 Topic
  35. All Pairs Shortest Paths
    6 Topics
  36. Backtracking
    General Method
  37. N-Queens Problem
  38. Sum of Subsets problem
  39. Graph Coloring
  40. Hamiltonian Cycles
  41. Branch and Bound
    2 Topics
  42. 0/1 Knapsack problem
    2 Topics
  43. NP-Complete and NP-Hard Problems
    1 Topic
Lesson 41, Topic 2
In Progress

Travelling Sales Person Problem

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We will be able to apply the branch-and-bound technique to instances of the traveling salesman problem if we come up with a reasonable lower bound on tour lengths. One very simple lower bound can be obtained by finding the smallest element in the intercity distance matrix D and multiplying it by the number of cities n. But there is a less obvious and more informative lower bound for instances with symmetric matrix D, which does not require a lot of work to compute. It is not difficult to show that we can compute a lower bound on the length l of any tour as follows. For each city i, 1 ≤ i ≤ n, find the sum si of the distances from city i to the two nearest cities; compute the sum s of these n numbers, divide the result by 2, and, if all the distances are integers, round up the result to the nearest integer:

lb = [s/2] —formula (a)

For example, for the instance in Figure a, formula (a) yields

lb = [[(1 + 3) + (3 + 6) + (1 + 2) + (3 + 4) + (2 + 3)]/2] = 14.

Moreover, for any subset of tours that must include particular edges of a given graph, we can modify lower bound (a) accordingly. For example, for all the Hamiltonian circuits of the graph in Figure a that must include edge (a, d), we get the following lower bound by summing up the lengths of the two shortest edges incident with each of the vertices, with the required inclusion of edges(a, d) and (d, a):

[[(1 + 5) + (3 + 6) + (1 + 2) + (3 + 5) + (2 + 3)]/2] = 16.

We now apply the branch-and-bound algorithm, with the bounding function given by formula (a), to find the shortest Hamiltonian circuit for the graph in Figure a. To reduce the amount of potential work, we take advantage of two observations. First, without loss of generality, we can consider only tours that start at a. Second, because our graph is undirected, we can generate only tours in which b is visited before c. In addition, after visiting n − 1 = 4 cities, a tour has no choice but to visit the remaining unvisited city and return to the starting one. The state-space tree tracing the algorithm’s application is given in Figure b.

KodNest Capture43
Figure (a) Weighted graph. (b) State-space tree of the branch-and-bound algorithm
to find a shortest Hamiltonian circuit in this graph. The list of vertices in
a node specifies a beginning part of the Hamiltonian circuits represented
by the node.

To reiterate the main point: these state-space tree techniques enable us to solve many large instances of difficult combinatorial problems. As a rule, however, it is virtually impossible to predict which instances will be solvable in a realistic amount of time and which will not.

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