This enables you to find which triangles correspond to which edges. Once you have this information you have to do two things. Compute an error metric for every edge in both directions. Most people only calculate the error metric of the edge in one direction, but I feel this can be something to avoid, as determining which point collapses into the other on the edge can be good for storage and for aesthetics. Well, there are many different ones.
Many people use more complex metrics which try to save the volume of the model etc.
The metric I use tries to avoid these things. Collapsing edges where there is a sharp angle between the edge and the other edges connected to it. To solve the first part of the problem, the metric takes the sum area of the 2 connected triangles into account. To solve the second part of the problem, the dot product between the normalised centroid vector of one triangle, and the normal of the other is used. To solve the third part of the error metric, we take the other edges connected to the point potentially being removed and take the sum of the normalised dot products between the vectors of the connected edge, and the edge we are testing.