  While quantum computing proposes promising solutions to computational
problems not accessible with classical approaches, due to current hardware
constraints, most quantum algorithms are not yet capable of computing systems
of practical relevance, and classical counterparts outperform them. To
practically benefit from quantum architecture, one has to identify problems and
algorithms with favorable scaling and improve on corresponding limitations
depending on available hardware. For this reason, we developed an algorithm
that solves integer linear programming problems, a classically NP-hard problem,
on a quantum annealer, and investigated problem and hardware-specific
limitations. This work presents the formalism of how to map ILP problems to the
annealing architectures, how to systematically improve computations utilizing
optimized anneal schedules, and models the anneal process through a simulation.
It illustrates the effects of decoherence and many body localization for the
minimum dominating set problem, and compares annealing results against
numerical simulations of the quantum architecture. We find that the algorithm
outperforms random guessing but is limited to small problems and that annealing
schedules can be adjusted to reduce the effects of decoherence. Simulations
qualitatively reproduce algorithmic improvements of the modified annealing
schedule, suggesting the improvements have origins from quantum effects.
