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An approximation algorithm is an algorithm that finds an approximate solution to a (typically NP-hard) problem. The quality of the algorithm is measured by how close to the actual optimum it performs. For example, it is a constant factor approximation algorithm if it always outputs a solution that is within a constant factor of the optimum. Hardness of approximation is one way to separate NP-hard problems.
5
votes
Approximating derivatives between gridpoints
Lagrange interpolation is the classic approach to this problem. In order for it to work, you'll need very precise function values, since the problem of numerical differentiation is ill-conditioned. …
2
votes
Greedy approach to 0-1 Knapsack problem in specific instances
As Frederico has already shown, this isn't enough to make the greedy heuristic immune to counterexamples.
It's worth mentioning that there is a simple dynamic programming algorithm for the knapsack …