Partially Matched Crossover

The idea of PMX is to copy a segment to preserve useful positional structure, and then repair conflicts systematically so every element appears exactly once.

Given parents $P_1,P_2$:

1. Choose two cut points $[c_1,c_2]$
2. Create child $C$ by copying segment $P_1[c_1:c_2]$ into $C[c_1:c_2]$
3. For each position $k\in [c_1,c_2]$, form a paired correspondence $$P_1[k]\leftrightarrow P_2[k]$$
4. Fill positions outside the segment using $P_2$, but if a value conflicts with the copied segment, resolve it by following the correspondence chain until a non-conflicting value is found.