  Integer programming (IP), as the name suggests is an integer-variable-based
approach commonly used to formulate real-world optimization problems with
constraints. Currently, quantum algorithms reformulate the IP into an
unconstrained form through the use of binary variables, which is an indirect
and resource-consuming way of solving it. We develop an algorithm that maps and
solves an IP problem in its original form to any quantum system possessing a
large number of accessible internal degrees of freedom that are controlled with
sufficient accuracy. This work leverages the principle of superposition to
solve the optimization problem. Using a single Rydberg atom as an example, we
associate the integer values to electronic states belonging to different
manifolds and implement a selective superposition of different states to solve
the full IP problem. The optimal solution is found within a few microseconds
for prototypical IP problems with up to eight variables and four constraints.
This also includes non-linear IP problems, which are usually harder to solve
with classical algorithms when compared to their linear counterparts. Our
algorithm for solving IP is benchmarked by a well-known classical algorithm
(branch and bound) in terms of the number of steps needed for convergence to
the solution. This approach carries the potential to improve the solutions
obtained for larger-size problems using hybrid quantum-classical algorithms.
