Adding o(n²) random edges to an αn-regular graph yields the same asymptotic minimum weights for spanning trees, shortest paths, and bipartite matchings as in the complete graph, scaled by α^{-1}.
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The effect of adding randomly weighted edges
Adding o(n²) random edges to an αn-regular graph yields the same asymptotic minimum weights for spanning trees, shortest paths, and bipartite matchings as in the complete graph, scaled by α^{-1}.