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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.

v[i] for m[i, j], and when m[i-1, j-w[i]] is out of range, we just give the value of m[i-1, j] to m[i, j].

From this perspective, we can program this method so that it runs recursively!

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That is to say, the program above computes more than expected because that the weight changes from 0 to W all the time.Thus, both versions of the problem are of similar difficulty.One theme in research literature is to identify what the "hard" instances of the knapsack problem look like, The goal in finding these "hard" instances is for their use in public key cryptography systems, such as the Merkle-Hellman knapsack cryptosystem.Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points.The students are asked to answer all of the questions to the best of their abilities.

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The knapsack problem is interesting from the perspective of computer science for many reasons: There is a link between the "decision" and "optimization" problems in that if there exists a polynomial algorithm that solves the "decision" problem, then one can find the maximum value for the optimization problem in polynomial time by applying this algorithm iteratively while increasing the value of k .

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