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Knapsack with Removability Hans-Joachim Bckenhauer, Jan Dreier, - PowerPoint PPT Presentation

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Threshold Behaviors in Advice Complexity Knapsack with Removability Hans-Joachim Bckenhauer, Jan Dreier, Fabian Frei , Peter Rossmanith August 28, 2020 Virtual satellite workshop of MFCS 2020 Online Problems and Advice Brief Recapitulation

  1. 1 Advice Bit: Lower Bound Hard Instance Family x 1 x 2 x 3 y 2 y 3 ψ 1 − ψ 2 + ε ψ 2 I 1 : ψ 1 − ψ 2 + ε ψ 2 1 − ψ 2 ψ 2 I 2 : ψ 1 − ψ 2 + ε ψ 2 − ε ψ 2 I 3 : ψ 1 − ψ 2 where ψ ≈ 0 . 78 is the positive root of 2(1 − x 2 ) = x Competitive Analysis Unique Optimum Second Best Competitivity ψ 2 ψ/ψ 2 I 1 : ψ 2(1 − ψ 2 ) + ε 1 / (2(1 − ψ 2 ) + ε ) I 2 : 1 I 3 : 1 ψ 1 /ψ

  2. 1 Advice Bit: Lower Bound Hard Instance Family x 1 x 2 x 3 y 2 y 3 ψ 1 − ψ 2 + ε ψ 2 I 1 : ψ 1 − ψ 2 + ε ψ 2 1 − ψ 2 ψ 2 I 2 : ψ 1 − ψ 2 + ε ψ 2 − ε ψ 2 I 3 : ψ 1 − ψ 2 where ψ ≈ 0 . 78 is the positive root of 2(1 − x 2 ) = x Competitive Analysis Unique Optimum Second Best Competitivity ψ 2 ψ/ψ 2 I 1 : ψ 2(1 − ψ 2 ) + ε 1 / (2(1 − ψ 2 ) + ε ) I 2 : 1 I 3 : 1 ψ 1 /ψ

  3. 1 Advice Bit: Lower Bound Hard Instance Family x 1 x 2 x 3 y 2 y 3 ψ 1 − ψ 2 + ε ψ 2 I 1 : ψ 1 − ψ 2 + ε ψ 2 1 − ψ 2 ψ 2 I 2 : ψ 1 − ψ 2 + ε ψ 2 − ε ψ 2 I 3 : ψ 1 − ψ 2 where ψ ≈ 0 . 78 is the positive root of 2(1 − x 2 ) = x Competitive Analysis Unique Optimum Second Best Competitivity ψ 2 ψ/ψ 2 I 1 : ψ 2(1 − ψ 2 ) + ε 1 / (2(1 − ψ 2 ) + ε ) I 2 : 1 I 3 : 1 ψ 1 /ψ

  4. 1 Advice Bit: Lower Bound Hard Instance Family x 1 x 2 x 3 y 2 y 3 ψ 1 − ψ 2 + ε ψ 2 I 1 : ψ 1 − ψ 2 + ε ψ 2 1 − ψ 2 ψ 2 I 2 : ψ 1 − ψ 2 + ε ψ 2 − ε ψ 2 I 3 : ψ 1 − ψ 2 where ψ ≈ 0 . 78 is the positive root of 2(1 − x 2 ) = x Competitive Analysis Unique Optimum Second Best Competitivity ψ 2 I 1 : ψ 1 /ψ 2(1 − ψ 2 ) + ε I 2 : 1 1 /ψ + ε I 3 : 1 ψ 1 /ψ

  5. 1 Advice Bit: Lower Bound Hard Instance Family x 1 x 2 x 3 y 2 y 3 ψ 1 − ψ 2 + ε ψ 2 I 1 : ψ 1 − ψ 2 + ε ψ 2 1 − ψ 2 ψ 2 I 2 : ψ 1 − ψ 2 + ε ψ 2 − ε ψ 2 I 3 : ψ 1 − ψ 2 where ψ ≈ 0 . 78 is the positive root of 2(1 − x 2 ) = x Competitive Analysis Unique Optimum Second Best Competitivity ψ 2 I 1 : ψ 1 /ψ 2(1 − ψ 2 ) + ε I 2 : 1 1 /ψ + ε I 3 : 1 ψ 1 /ψ

  6. 1 Advice Bit: Lower Bound Hard Instance Family x 1 x 2 x 3 y 2 y 3 ψ 1 − ψ 2 + ε ψ 2 I 1 : ψ 1 − ψ 2 + ε ψ 2 1 − ψ 2 ψ 2 I 2 : ψ 1 − ψ 2 + ε ψ 2 − ε ψ 2 I 3 : ψ 1 − ψ 2 where ψ ≈ 0 . 78 is the positive root of 2(1 − x 2 ) = x Competitive Analysis Unique Optimum Second Best Competitivity ψ 2 I 1 : ψ 1 /ψ ≈ 1 . 28 2(1 − ψ 2 ) + ε I 2 : 1 vs. I 3 : 1 ψ 1 /ψ 10 / 7 ≈ 1 . 43

  7. 1 Advice Bit Upper Bound

  8. 1 Advice Bit: Improved Upper Bound High-Level Outline Only • Divide items into 5 size classes • Two different packing strategies • Use advice bit to indicate which one is better √ • Yields a 2-competitive algorithm little large tiny small medium big huge 0 1

  9. Near-Optimality Using Constant Advice

  10. Near-Optimality with Constant Advice Algorithm Requirements • Takes parameter ε > 0 • Is (1 + ε )-competitive • Uses only constant advice

  11. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  12. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  13. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  14. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  15. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  16. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  17. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  18. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  19. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  20. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · (1 − ε ) E (1 − ε ) E − 2 (1 − ε ) 2 0 1

  21. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · ε (1 − ε ) E − 2 (1 − ε ) 2 0 1

  22. Near-Optimality with Constant Advice Algorithm Outline • Given ε > 0 • Let E = log 1 − ε ε • Divide items into E + 1 size classes as shown below (1 − ε ) E − 1 (1 − ε ) 3 1 − ε · · · ε (1 − ε ) E − 2 (1 − ε ) 2 0 1 Big items Small items

  23. Near-Optimality with Constant Advice What does the advice encode? • Oracle fixes arbitrary optimal solution S • Let u 1 , . . . , u k denote big items of S in appearance order • Let c i denote size class of u i • The advice encodes ( c 1 , . . . , c k )

  24. Near-Optimality with Constant Advice Only constant advice • 2 k ⌈ log 2 ( E + 1) ⌉ bits suffice to encode ( c 1 , . . . , c k ) – There are E classes for big items – Thus one class indicated by ⌈ log 2 ( E + 1) ⌉ bits – A self-delimiting encoding: 2 ⌈ log 2 ( E + 1) ⌉ bits

  25. Near-Optimality with Constant Advice Only constant advice • 2 k ⌈ log 2 ( E + 1) ⌉ bits suffice to encode ( c 1 , . . . , c k ) – There are E classes for big items – Thus one class indicated by ⌈ log 2 ( E + 1) ⌉ bits – A self-delimiting encoding: 2 ⌈ log 2 ( E + 1) ⌉ bits • This is constant advice: – The number k of big elements in S is bounded by 1 /ε – Recall that E = log 1 − ε ( ε )

  26. Near-Optimality with Constant Advice Algorithm Procedure • Big items: Packed into k virtual slots – Slot i accommodates items from class c i exclusively – In the beginning, the slots are empty – Each slot is filled with exactly one big item – The slots are filled strictly in their order – Items in filled slots replaced by smaller ones if possible

  27. Near-Optimality with Constant Advice Algorithm Procedure • Big items: Packed into k virtual slots – Slot i accommodates items from class c i exclusively – In the beginning, the slots are empty – Each slot is filled with exactly one big item – The slots are filled strictly in their order – Items in filled slots replaced by smaller ones if possible • Small items: Packed greedily – Removed one by one whenever needed to pack a big one

  28. Example Execution of the Algorithm

  29. Execution Example Optimal Solution:

  30. Execution Example 0 1 Optimal Solution:

  31. Execution Example Optimal Solution: 2 1 3 2

  32. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice:

  33. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2

  34. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output:

  35. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output:

  36. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output:

  37. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output:

  38. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2

  39. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2

  40. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1

  41. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1

  42. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3

  43. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3

  44. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3

  45. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3

  46. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3

  47. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3

  48. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  49. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  50. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  51. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  52. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  53. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  54. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  55. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  56. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  57. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  58. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  59. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  60. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

  61. Execution Example Optimal Solution: 2 1 3 2 ( c 1 , c 2 , c 3 , c 4 ) = (2 , 1 , 3 , 2) Advice: Input: 1 2 1 3 2 3 2 2 1 1 3 2 Output: 2 1 3 2

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