Consider the following maximization version of the 3-Dimensional Matching Problem. Given disjoint sets X, Y, and Z, and given a set T ⊆ X x Y x Z of ordered triples, a subset M c T is a 3-dimensional matching if each element of X ∩ Y U Z is contained in at most one of these triples. The Maximum 3-Dimensional Matching Problem is to find a 3-dimensional matching M of maximum size. (The size of the matching, as usual, is the number of triples it contains. You may assume |X| = | Y| = |Z| if you want.)
Give a polynomial-time algorithm that finds a 3-dimensional matching of size at least 3 times the maximum possible size.
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