1. Robbing a House with Greedy Algorithms
Casey O’Brien

2. Knapsack Problem

4. Your Knapsack
Max Weight: 10 pounds

5. Item

6. Item
Value 13 5 3 6 9

7. Item
Value
Weight 13 8 5 3
0.6 6
1.5 9

8. Goal: Maximize total value
Constraint: Total weight cannot exceed 10 pounds

9. Item
Value
Weight 13 8 5 3
0.6 6
1.5 9

10. Item
Value
Weight
Value/Weight 13 8
1.6 5 1 3
0.6 6
1.5 4 9

11. Play The Robber!

12. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

13. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

14. Item
Value
Weight
Value/Weight
Knapsack 12 8
1.6 5 1 3
0.6 6
1.5 4 9

15. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

16. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

17. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

18. Final Knapsack:
Final Value: $14

19. Play The Robber!

20. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

21. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

22. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

23. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

24. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

25. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

26. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

27. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

28. Final Knapsack:
Final Value: $19

29. Final Knapsack:
Final Value: $16

30. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

31. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

32. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

33. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

34. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

35. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

36. Final Knapsack:
Final Value: $14

37. Final Knapsack:
Final Value: $20

38. Final Knapsack:
Final Value: $17

39. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

40. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

41. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

42. Final Knapsack:
Final Value: $15

43. Final Knapsack:
Final Value: $18

44. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

45. Greedy Algorithms

46. Greedy by Largest Value

47. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

48. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

49. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

50. Final Knapsack:
Final Value: $19

51. Greedy by Smallest Weight

52. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

53. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

54. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

55. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

56. Final Knapsack:
Final Value: $18

57. Greedy by Largest Ratio

58. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

59. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

60. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

61. Item
Value
Weight
Value/Weight
Knapsack 13 8
1.6 5 1 3
0.6 6
1.5 4 9

62. Final Knapsack:
Final Value: $18

63. Greedy Algorithms do not guarantee optimal solution
Problem:
Greedy Algorithms do not guarantee optimal solution

64. Exhaustive Enumeration

65. Exhaustive Enumeration
Try All Possibilities

66. How Many Possibilities?

75. Final Knapsack:
Final Value: $20

76. Problem:
Takes Too Long!

77. 40 Items

78. ~1 Trillion Possibilities
40 Items
~1 Trillion Possibilities

79. ~1 Trillion Possibilities
40 Items
~1 Trillion Possibilities
~35,000 Years to Compute

80. What Can We Do?

81. Settle For Less Than Optimal

82. Recall Greedy Algorithms:
By Value: $19
By Weight: $18
By Ratio: $18
Optimal: $20

83. Moral of the Story:
We can use Greedy Algorithms to approximate solutions to the knapsack problem quickly.