# Dating problem cryptography

A similar dynamic programming solution for the 0/1 knapsack problem also runs in pseudo-polynomial time. From Definition A, we can know that there is no need for computing all the weights when the number of items and the items themselves that we chose are fixed.That is to say, the program above computes more than expected because that the weight changes from 0 to W all the time.The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography, applied mathematics, and daily fantasy sports.The knapsack problem has been studied for more than a century, with early works dating as far back as 1897.To be exact, the knapsack problem has a fully polynomial time approximation scheme (FPTAS)..It then proceeds to insert them into the sack, starting with as many copies as possible of the first kind of item until there is no longer space in the sack for more.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).As for most NP-complete problems, it may be enough to find workable solutions even if they are not optimal.

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.

All we need to do is to compare m[i-1, j] and m[i-1, j-w[i 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!

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 .

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 .

||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.All we need to do is to compare m[i-1, j] and m[i-1, j-w[i]] 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!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 .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 .

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