2. 1 Combinatorial Dominance Analysis The Knapsack Problem Keywords: Combinatorial Dominance (CD) Domination number/ratio (domn, domr) Knapsack (KP) Incremental Insertion (II) Local Exchange (LE) PTAS Optimal Head - Greedy Tail (GRT) Presented by: Yochai Twitto

3. 2 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. The Knapsack Problem simple Algorithms & Analysis Incremental Insertion Local Exchange PTASing “ Optimal head - greedy tail ” algorithm Summary

4. 3 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. The Knapsack Problem simple Algorithms & Analysis Incremental Insertion Local Exchange PTASing “ Optimal head - greedy tail ” algorithm Summary

5. 4 Background NP complexity class. AA and quality of approximations. The classical approximation ratio analysis.

6. 5 NP If P ≠ NP, then finding the optimum of NP-hard problem is difficult. If P = NP, P would encompass the NP and NP-Complete areas.

7. 6 Approximations So we are satisfied with an approximate solution. Question: How can we measure the solution quality ? Solutions quality line OPT Infeasible Near optimal

8. 7 Solution Quality Most of the time, naturally derived from the problem definition. If not, it should be given as external information.

9. 8 The classical Approximation Ratio (For maximization problem) Assume 0 ≤ β ≤ 1. A.r. ≥ β if the solution quality is greater than β·OPT Solutions quality line OPT Infeasible Near optimal ½ OPT

10. 9 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. The Knapsack Problem simple Algorithms & Analysis Incremental Insertion Local Exchange PTASing “ Optimal head - greedy tail ” algorithm Summary

11. 10 Combinatorial Dominance What is a “ combinatorial dominance guarantee ” ? Why do we need such guarantees ? Definitions and notations.

12. 11 What is a “ combinatorial dominance guarantee ” ? A letter of reference: “ She is half as good as I am, but I am the best in the world …” “ she finished first in my class of 75 students …” The former is akin to an approximation ratio. The latter to combinatorial dominance guarantee.

13. 12 What is a “ combinatorial dominance guarantee ” ? (cont.) We can ask: Is the returned solution guaranteed to be always in the top O(n) best solutions ? Solutions quality line OPT Infeasible Near optimal top O(n)

14. 13 Why do we need that ? Assume an problem for which all solutions are at least a half as good as optimal solution. Then, 2-factor approximating the problem is meaningless.

15. 14 Corollary The approximation ratio analysis gives us only a partial insight of the performance of the algorithm. Dominance analysis makes the picture fuller.

16. 15 Definitions & Notations Domination number: domn Domination ratio: domr

17. 16 Domination Number: domn Let P be a CO problem. Let A be an approximation for P. For an instance I of P, the domination number domn( I, A ) of A on I is the number of feasible solutions of I that are not better than the solution found by A.

18. 17 domn (example) STSP on 5 vertices. There exist 12 tours If A returns a tour of length 7 then domn( I, A ) = 8 4, 5, 5, 6, 7, 9, 9, 11, 11, 12, 14, 14 (tours lengths)

19. 18 Domination Number: domn Let P be a CO problem. Let A be an approximation for P. minimum For any size n of P, the domination number domn( P, n, A ) of an approximation A for P is the minimum of domn( I, A ) over all instances I of P of size n.

20. 19 Domination Ratio: domr Let P be a CO problem. Let A be an approximation for P. sol( I )feasible Denote by sol( I ) the number of all feasible solutions of I. minimum For any size n of P, the domination ratio domn( P, n, A ) of an approximation A for P is the minimum of domn( I, A ) / sol( I ) taken over all instances I of P of size n.

21. 20 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. The Knapsack Problem simple Algorithms & Analysis Incremental Insertion Local Exchange PTASing “ Optimal head - greedy tail ” algorithm Summary

22. 21 The Knapsack Problem Instance: Multiset of integers Capacity Find:

23. 22 Simple Algorithms & Analysis Incremental Insertion (II) Arbitrary order Increasing order Decreasing order (Greedy) Local Exchange (LE) PTASing “ Optimal Head – Greedy Tail ” (GRT)

24. 23 II – Arbitrary Order Go over the elements (arbitrary order) Insert an element if the capacity not exceeded Theorem:

25. 24 Proof Suppose the weights are Let be any locally optimal solution We may assume Otherwise, is optimal 

26. 25 Proof (cont.) Let be the largest index of a weight not belonging to  Since is locally optimal

27. 26 Proof (cont.) Denote by the interval For any solution not containing Either Or That is, the number of solutions with total weight in is at most

28. 27 Proof (cont.) Solutions of weight at least are infeasible. Solution weighted not more than are not better than 

29. 28 Proof (cont.) Blackball instance: II can lead to Which is locally optimal blackball

30. 29 Proof (cont.) Taking the first item and omitting at least one of the rest is better. Hence And we finished...

31. 30 II – Increasing Order No Gain! That was our blackball … In the previous proof.

32. 31 II - Decreasing Order (Greedy) No drastic gain! Blackball instance B: blackball

33. 32 II - Decreasing Order (Greedy) Greedy(B)  Weight: Any solution taking Exactly two elements from Any of the last elements is better!

34. 33 II - Decreasing Order (Greedy)

35. 34 Simple Algorithms & Analysis Incremental Insertion (II) Arbitrary order Increasing order Decreasing order (Greedy) Local Exchange (LE) PTASing “ Optimal Head – Greedy Tail ” (GRT)

36. 35 Local Exchange (LE) Assume is a solution Allowed operations: Insert a new element x to Exchange x by y x belongs to y not belongs to x < y

37. 36 Local Exchange Theorem:

38. 37 Proof Suppose the weights are Let be any locally optimal solution We may assume Otherwise, is optimal 

39. 38 Proof (cont.) Let be the largest index of a weight not belonging to  Since is locally optimal

40. 39 Proof (cont.) Denote by the interval For any solution not containing Either Or That is, the number of solutions with total weight in is at most And there are at least outside

41. 40 Proof (cont.) Let be the number of items belonging to among the first k -1 items Let be the number of items not belonging to among the first k -1 items How many solution pairs are of weight not belonging to ?

42. 41 Proof (cont.) We saw that All solutions obtained by dispensing of some items from And the one obtained from them by adjoining the ’ th item not belong to the interval

43. 42 Proof (cont.) So For each of the solutions obtained from by adjoining one of the items of Both the obtained solution And the one obtain by adjoining it the ’ th item not belong to the interval

44. 43 Proof (cont.) So Since our solution can not be improved by local exchange Each of the n-k solutions obtained by removing one of the last n-k items not belong to the interval Adding each of them the ’ th item we get infeasible solutions

45. 44 Proof (cont.) So

46. 45 Proof (cont.) Blackball instance: LE can lead to Which is locally optimal blackball

47. 46 Proof (cont.) Taking the first item and omitting at least two of the rest is better. Hence: And we finished... b(n)

48. 47 Simple Algorithms & Analysis Incremental Insertion (II) Arbitrary order Increasing order Decreasing order (Greedy) Local Exchange (LE) PTASing “ Optimal Head – Greedy Tail ” (GRT)

49. 48 PTASing There exist a PTAS for Knapsack That is, it is possible to approximate the optimal solution to within any factor c >1 In time polynomial in n and 1/(c -1) We ’ ll see

50. 49 Theorem 1 Let be an instance of KP Denote the weight of optimal solution by Assume H is a factor-c approximation for KP Then

51. 50 Proof Assume that the elements of optimal solution are labeled such that Let ’ be the smallest integer such that

52. 51 Proof (cont.) Let Observe that

53. 52 Proof (cont.) Also note that Since The weight of every element of is not more than the weight of any element of is a c -approximated solution to

54. 53 Proof (cont.) Let is minimized for Since

55. 54 Proof (cont.) Note that our solution dominate united with any of the non-empty subsets of Since they are not feasible Since is optimal

56. 55 Proof (cont.) Note that our solution dominate all subset of Since the weight of each is not more than And our solution weight is at least

57. 56 Proof (cont.) Summing both terms, the number of solution dominated is Minimizing the left-hand term we get the result.

58. 57 Theorem 2 For every c >1 there exist a KP instance and a solution thereof of total weight dominating only solution.

59. 58 Proof Blackball instance: can return a solution consisting of items blackball

60. 59 Proof (cont.) Such solution dominates all solutions consisting of up to item It also dominates all infeasible solution i.e: solution consisting of more than items. Those are the only solution it dominates

61. 60 Proof (cont.) Hence Employing Stirling ’ s formula we obtain the theorem

62. 61 Simple Algorithms & Analysis Incremental Insertion (II) Arbitrary order Increasing order Decreasing order (Greedy) Local Exchange (LE) PTASing “ Optimal Head – Greedy Tail ” (GRT)

63. 62 Optimal Head – Greedy Tail

64. 63 Optimal Head – Greedy Tail

65. 64 Optimal Head – Greedy Tail

66. 65 Optimal Head – Greedy Tail

67. 66 Optimal Head – Greedy Tail

68. 67 Optimal Head – Greedy Tail

69. 68 Optimal Head – Greedy Tail

70. 69 Optimal Head – Greedy Tail

71. 70 Optimal Head – Greedy Tail

72. 71 Summary

73. 72 Combinatorial Dominance Analysis The Knapsack Problem