  Combinatorial optimization problems, such as the Asymmetric Traveling
Salesman Problem (ATSP), find applications across various domains including
logistics, genome sequencing, and robotics. Despite their extensive
applications, there have not been significant advancements in deriving optimal
solutions for these problems. The lack of theoretical understanding owing to
the complex structure of these problems has hindered the development of
sophisticated algorithms. This paper proposes an unconventional approach by
translating the ATSP into the complex domain, revealing an intrinsic modular
nature of the problem. Furthermore, we have exploited modularity conditions to
gain deeper insights into both unconstrained and constrained optimal solutions.
The theoretical framework laid out in this paper can lead to important results
at the intersection of combinatorial optimization and number theory.
