Consider the setting of randomly weighted graphs, namely, graphs whose edge weights are chosen independently according to probability distributions with finite support over the non-negative reals. Under this setting, weighted graph properties such as the diameter, theradius (with respect to a designated vertex), and the weight of a minimum spanning tree become random variables and we are interested in computing their expectation. Unfortunately, this turns out to be #P-hard. In this paper, we define a family of weighted graph properties (that includes the above three) and show that for each property in this family, the problem of computing the k th moment (and in particular, the expectation) of the corresponding random variable admits a fully polynomial-time randomized approximation scheme (FPRAS) for every fixed k.
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