Step 1: Understanding anaerobic decarboxylation.
In anaerobic respiration, pyruvate is decarboxylated to acetaldehyde by the enzyme pyruvate decarboxylase.
Step 2: Required coenzyme and cofactor.
This enzyme requires thiamine pyrophosphate (TPP) as a coenzyme and Zn$^{++}$ as a cofactor for its activity.
Step 3: Conclusion.
Thus, the correct pair is Thiamine pyrophosphate and Zn$^{++$}.