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2 well known approximations (MAX 3DM)
When phrased as the optimization problem here and here, there are 2 well know approaches. The first gets a round of sups that is within 1/3 of the best possible round, the second gets a round of sups that is within 2/3 + ε of the best possible round.

from Maximum bounded 3-dimensional matching is MAX SNP-complete Viggo Kann Royal Institute of Technology, Stockholm

from here