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Preferably, however, the approximation comes with a guarantee on the difference between the value of the solution found and the value of the optimal solution.

As with many useful but computationally complex algorithms, there has been substantial research on creating and analyzing algorithms that approximate a solution.

For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve a maximum possible score of 100 points.

However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices.

Another algorithm for 0-1 knapsack, discovered in 1974 and sometimes called "meet-in-the-middle" due to parallels to a similarly named algorithm in cryptography, is exponential in the number of different items but may be preferable to the DP algorithm when input: a set of items with weights and values output: the greatest combined value of a subset partition the set into two sets A and B of approximately equal size compute the weights and values of all subsets of each set for each subset of A find the subset of B of greatest value such that the combined weight is less than W keep track of the greatest combined value seen so far space, and efficient implementations of step 3 (for instance, sorting the subsets of B by weight, discarding subsets of B which weigh more than other subsets of B of greater or equal value, and using binary search to find the best match) result in a runtime of ), at the cost of using exponential rather than constant space (see also baby-step giant-step).

On the other hand, if an algorithm finds the optimal value of the optimization problem in polynomial time, then the decision problem can be solved in polynomial time by comparing the value of the solution output by this algorithm with the value of k .Of the possible subsets of problems whose total point values add up to 100, a knapsack algorithm would determine which subset gives each student the highest possible score. Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity.The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number One example of the unbounded knapsack problem is given using the figure shown at the beginning of this article and the text "if any number of each box is available" in the caption of that figure.Provided that there is an unlimited supply of each kind of item, if .However, for the bounded problem, where the supply of each kind of item is limited, the algorithm may be far from optimal.