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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.

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Compute the average path weights of paths with the same path length in a directed acyclic gr...

No need for an external graph, you can use the following recurrence (for edge and vertex weights) which is very similar to yours: $n_v(l)$ number of paths of length $l$ that end at $v$ $w_v(l)$ sum o …
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