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The tourists want to pack n items into two infinite-sized knapsacks

INSTRUCTIONS TO CANDIDATES

The tourists want to pack n items into two infinite-sized knapsacks (i.e. each knapsack can contain an unlimited number of items of any size). The size of each item is a positive real number.

It is known that the maximal size of any single item is 3, and the total size of all items together is 15. The goal is to minimize the maximum total size of the items packed into any of the two knapsacks.

The tourists use the following greedy algorithm. The items are ordered in an arbitrary order. The item under consideration is placed into the knapsack with the smallest total size of items in it. Show that at the end of the algorithm run, the total size of items in the knapsack with the largest total size of its items is at most 1.2 times the optimum.

Example: assume that n = 7 and that the items have sizes 1, 2, 2, 3, 1, 3, 3 (in this order). The result of the tourists' algorithm is:

Knapsack 1: {1, 2, 1,3}, the total size is 7.

Knapsack 2: {2, 3, 3}, the total size is 8.

The largest total size is 8.

(5/5)

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