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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 43, Topic 1
In Progress

Basic Concepts

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The theory of NP-completeness which we present here does not provide a
method of obtaining polynomial time algorithms for problems in the second
group. Nor does it say that algorithms of this complexity do not exist.
Instead,what we do is show that many of the problems for which there are no known polynomial time algorithms are computationally related. In fact,
we establish two classes of problems. These are given the names NP-hard
and NP-complete. A problem that is NP-complete has the property that
it can be solved in polynomial time if and only if all other NP-complete
problems can also be solved in polynomial time. If an NP-hard problem
can be solved in polynomial time,then all NP-complete problems can be
solved in polynomial time. All NP-complete problems are NP-hard, but
some NP-hard problems are not known to be NP-complete.

Although one can define many distinct problem classes having the
properties stated above for the NP -hard and NP -complete classes,the classes we study are related to nondeterministic computations(to be defined later). The relationship of these classes to nondeterministic computations together with the apparent power of nondeterminism leads to the intuitive (though as yet unproved)conclusion that no NP-complete or NP-hard problem is polynomially solvable.

We see that the class of NP-hard problems(and the subclass of NV complete problems)is very rich as it contains many interesting problems from
a wide variety of disciplines. First,we formalize the preceding discussion of
the classes.

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