  While lateral interaction models for reactions at surfaces have steadily
gained popularity and grown in terms of complexity, their use in chemical
kinetics has been impeded by the low performance of current KMC algorithms. The
origins of the additional computational cost in KMC simulations with lateral
interactions are traced back to the more elaborate Cluster Expansion
Hamiltonian, the more extensive rate updating, and to the impracticality of
rate-catalog-based algorithms for interacting adsorbate systems. Favoring
instead site-based algorithms, we propose three ways to reduce the cost of KMC
simulations: 1. Represent the lattice energy by a smaller Supercluster
Hamiltonian without loss of accuracy, 2. employing Subtraction Schemes for
updating key quantities in the simulation that undergo only small, local
changes during a reaction event, and 3. applying efficient search algorithms
from a set of established methods (Supersite Approach). The resulting algorithm
is fixed-cost with respect to the number of lattice sites for practical lattice
sizes and scales with the square of the range of lateral interactions. The
overall added cost of including a complex lateral interaction model amounts to
less than a factor 3. Practical issues in implementation due to finite
numerical accuracy are discussed in detail, and further suggestions for
treating long-range lateral interactions are made. We conclude that, while KMC
simulations with complex lateral interaction models are challenging, these
challenges can be overcome by modifying the established Variable Step Size
Method by employing the Supercluster, Subtraction and Supersite algorithms
(SSS-VSSM).
