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Optimal Knapsack Subset Problem

Given a number B and a collection of items A, each with a value and size, find a subset A' maximizing value within size limit B. Example provided.

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Optimal Knapsack Subset Problem

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  1. Knapsack: Given a number B and a collection of n items A = {x[0], x[1], x[2],..., x[n]}, each with some positive integer value v(x[i]) and some positive integer size s(x[i]), find a subset A' of A such that the sum of the values of items in A' is greatest over all subsets of A whose sum of sizes is no greater than B. Example: B=27 A = { x[0]...x[4] } v(x[0]) = 23, s(x[0]) = 11 v(x[1]) = 17, s(x[1]) = 7 v(x[2]) = 21, s(x[2]) = 16 v(x[3]) = 26, s(x[3]) = 10 v(x[4]) = 19, s(x[4]) = 9 Solution: { x[1], x[3], x[4] }

  2. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27

  3. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16

  4. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] is in the knapsack 40 : <21,6>, <26,10>, <19,9> : 9

  5. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] is in the knapsack 40 : <21,6>, <26,10>, <19,9> : 9 . x[2] is in the knapsack 61 : <26,10>, <19,9> : 3

  6. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] is in the knapsack 40 : <21,6>, <26,10>, <19,9> : 9 . x[2] 61 : <26,10>, <19,9> : 3 Best value so far: 61

  7. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] is in the knapsack 40 : <21,6>, <26,10>, <19,9> : 9 . x[2] not in the knapsack . x[2] 61 : <26,10>, <19,9> : 3 40 : <26,10>, <19,9> : 9 Best value so far: 61

  8. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] is in the knapsack 40 : <21,6>, <26,10>, <19,9> : 9 . x[2] not in the knapsack . x[2] 61 : <26,10>, <19,9> : 3 40 : <26,10>, <19,9> : 9 Best value so far: 61 . x[3] not in the knapsack 40 : <19,9> : 9

  9. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] is in the knapsack 40 : <21,6>, <26,10>, <19,9> : 9 . x[2] not in the knapsack . x[2] 61 : <26,10>, <19,9> : 3 40 : <26,10>, <19,9> : 9 Best value so far: 61 . x[3] not in the knapsack 40 : <19,9> : 9 . x[4] in the knapsack 59 : : 0

  10. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] 40 : <21,6>, <26,10>, <19,9> : 9 . x[2] 61 : <26,10>, <19,9> : 3 Best value so far: 61

  11. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack . x[1] 40 : <21,6>, <26,10>, <19,9> : 9 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] 61 : <26,10>, <19,9> : 3 Best value so far: 61

  12. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack . x[1] 40 : <21,6>, <26,10>, <19,9> : 9 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] . x[2] in the knapsack 61 : <26,10>, <19,9> : 3 44 : <26,10>, <19,9> : 10 Best value so far: 61

  13. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack . x[1] 40 : <21,6>, <26,10>, <19,9> : 9 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] . x[2] in the knapsack 61 : <26,10>, <19,9> : 3 44 : <26,10>, <19,9> : 10 Best value so far: 61 . x[3] in the knapsack 70 : <19,9> : 0

  14. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack . x[1] 40 : <21,6>, <26,10>, <19,9> : 9 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] . x[2] in the knapsack 61 : <26,10>, <19,9> : 3 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  15. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  16. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  17. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] not in the knapsack . x[3] in the knapsack 70 : <19,9> : 0 44 : <19,9> : 10 Best value so far: 70

  18. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] not in the knapsack . x[3] in the knapsack 70 : <19,9> : 0 44 : <19,9> : 10 Cannot beat 70 Best value so far: 70

  19. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  20. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] is not in the knapsack . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 23 : <26,10>, <19,9> : 16 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  21. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] is not in the knapsack . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 23 : <26,10>, <19,9> : 16 . x[3] in the knapsack Cannot beat 70 70 : <19,9> : 0 Best value so far: 70

  22. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  23. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 16 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  24. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  25. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 38 : <26,10>, <19,9> : 14 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  26. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 38 : <26,10>, <19,9> : 14 . x[3] in the knapsack . x[3] in the knapsack 70 : <19,9> : 0 64 : <19,9> : 4 Best value so far: 70 Cannot beat 70

  27. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 38 : <26,10>, <19,9> : 14 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  28. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 38 : <26,10>, <19,9> : 14 . x[3] in the knapsack . x[3] not in the knapsack 70 : <19,9> : 0 38 : <19,9> : 14 Best value so far: 70 Cannot beat 70

  29. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  30. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack . x[2] not in the knapsack 44 : <26,10>, <19,9> : 10 17 : <26,10>, <19,9> : 20 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  31. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] in knapsack . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 17 : <21,6>, <26,10>, <19,9> : 20 . x[2] in the knapsack . x[2] not in the knapsack 44 : <26,10>, <19,9> : 10 17 : <26,10>, <19,9> : 20 . x[3] in the knapsack Cannot beat 70 70 : <19,9> : 0 Best value so far: 70

  32. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] not in the knapsack 23 : <21,6>, <26,10>, <19,9> : 16 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  33. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] not in the knapsack . x[1] not in knapsack 23 : <21,6>, <26,10>, <19,9> : 16 0 : <21,6>, <26,10>, <19,9> : 27 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  34. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] not in the knapsack . x[1] not in knapsack 23 : <21,6>, <26,10>, <19,9> : 16 0 : <21,6>, <26,10>, <19,9> : 27 Cannot beat 70 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Best value so far: 70

  35. 0 : <23,11>, <17,7>, <21,16>, <26,10>, <19,9> : 27 Assume x[0] is in the knapsack . x[0] not in knapsack 23 : <17,7>, <21,6>, <26,10>, <19,9> : 16 0 : <17,7>, ..., <19,9> : 27 . x[1] not in the knapsack . x[1] not in knapsack 23 : <21,6>, <26,10>, <19,9> : 16 0 : <21,6>, <26,10>, <19,9> : 27 . x[2] in the knapsack 44 : <26,10>, <19,9> : 10 . x[3] in the knapsack 70 : <19,9> : 0 Solution: 70

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