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The multi-type Moran process is an evolutionary process on a connected graph$G$ in which each vertex has one of $k$ types and, in each step, a vertex $v$is chosen to reproduce its type to one of its neighbours. The probability of avertex $v$ being chosen for reproduction is proportional to the fitness of thetype of $v$. So far, the literature was almost solely concerned with the$2$-type Moran process in which each vertex is either healthy (type $0$) or amutant (type $1$), and the main problem of interest has been the (approximate)computation of the so-called fixation probability, i.e., the probability thateventually all vertices are mutants.  In this work we initiate the study of approximating fixation probabilities inthe multi-type Moran process on general graphs. Our main result is an FPTRAS(fixed-parameter tractable randomised approximation scheme) for computing thefixation probability of the dominant mutation; the parameter is the number oftypes and their fitnesses. In the course of our studies we also provide novelupper bounds on the expected absorption time, i.e., the time that it takes themulti-type Moran process to reach a state in which each vertex has the sametype.

Parameterised Approximation of the Fixation Probability of the Dominant Mutation in the Multi-Type Moran Process