3-dimensional matching approximation algorithm (implementation details)

We don’t need the maximum matching, just one of cardinality $2$ if it exists. Scan $S_r$ looking for a triple $T$ such that $lvert T cap {a, b, c}rvert = 1$. If no such $T$ exists, then the maximum cardinality of a matching is clearly $1$. Otherwise, assume without loss of generality that $T = {a, x, y}$. Scan $S_r$ again looking for a triple $U$ such that $T cap U = varnothing$. If no such $U$ exists, then every triple intersects both ${a, b, c}$ and $T$. For every $U in S_r$, we have ${a} subseteq U$ or ${b, x} subseteq U$ or ${b, y} subseteq U$ or ${c, x} subseteq U$ or ${c, y} subseteq U$. There are several possibilities for a cardinality-$2$ matching ${T_1, T_2}$. There’s an easy test for those of type ${b, x} subseteq T_1$ and ${c, y} subseteq T_2$. Similarly, ${b, y}$ and ${c, x}$. The only other types are the four like ${a} subseteq T_1$ and ${b, x} subseteq T_2$. To test for those, gather all triples of the form ${b, x, z} in S_r$ and discard the ones in excess of three. Try each of the remainder against all possibilities containing $a$. The three ${b, x, z}$ candidates suffice because no triple containing neither $b$ nor $x$ intersects all three.