This is a well-known problem with solutions in the literature e.g.
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:) Often given multi-dimensional data that describe data points, you define a similarity or "kernel" between points. dot product after you normalize by standard deviation in each dimension for example.
Or it could be a Gaussian kernel e^((-d^2)/y) where d is the dot-product between points and y is a constant bandwidth parameter. if certain dimensions are categorical then you could the one-dimensional dot-product to be 1 if the categorical variables agree, otherwise 0.
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Then you can form the overall dot-product from the multi-dimensional data after normalizing each dimension by its standard deviation.
The point is, once you form a similarity or kernel between points, then you can define a weighted bipartite graph where the weight of an edge is equal to the similarity/kernel between points, and your problem is to find a maximum weight matching.