  Fully pairing all elements of a set while attempting to maximize the total
benefit is a combinatorically difficult problem. Such pairing problems
naturally appear in various situations in science, technology, economics, and
other fields. In our previous study, we proposed an efficient method to infer
the underlying compatibilities among the entities, under the constraint that
only the total compatibility is observable. Furthermore, by transforming the
pairing problem into a traveling salesman problem with a multi-layer
architecture, a pairing optimization algorithm was successfully demonstrated to
derive a high-total-compatibility pairing. However, there is substantial room
for further performance enhancement by further exploiting the underlying
mathematical properties. In this study, we prove the existence of algebraic
structures in the pairing problem. We transform the initially estimated
compatibility information into an equivalent form where the variance of the
individual compatibilities is minimized. We then demonstrate that the total
compatibility obtained when using the heuristic pairing algorithm on the
transformed problem is significantly higher compared to the previous method.
With this improved perspective on the pairing problem using fundamental
mathematical properties, we can contribute to practical applications such as
wireless communications beyond 5G, where efficient pairing is of critical
importance.
