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 a vertex $v$ being chosen for reproduction is proportional to the fitness of the type 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 a mutant (type $1$), and the main problem of interest has been the (approximate) computation of the so-called fixation probability, i.e., the probability that eventually all vertices are mutants. In this work we initiate the study of approximating fixation probabilities in the multi-type Moran process on general graphs. Our main result is an FPTRAS (fixed-parameter tractable randomised approximation scheme) for computing the fixation probability of the dominant mutation; the parameter is the number of types and their fitnesses. In the course of our studies we also provide novel upper bounds on the expected absorption time, i.e., the time that it takes the multi-type Moran process to reach a state in which each vertex has the same type.

翻译：多类型莫兰过程是一种在连通图 $G$ 上演化的进化过程，其中每个顶点具有 $k$ 种类型之一，每一步选择一个顶点 $v$ 将其类型复制给一个邻居。顶点 $v$ 被选中进行繁殖的概率与其类型的适应度成正比。迄今为止，文献几乎仅关注 $2$ 类型莫兰过程，其中每个顶点要么是健康类型（类型 $0$），要么是突变类型（类型 $1$），主要问题在于（近似）计算所谓的固定概率，即最终所有顶点均为突变类型的概率。本研究首次探讨了一般图上多类型莫兰过程中固定概率的近似问题。我们的主要成果是用于计算主导突变固定概率的FPTRAS（固定参数可追踪随机近似方案）；参数为类型数量及其适应度。在研究过程中，我们还提供了期望吸收时间的新上界，即多类型莫兰过程达到每个顶点具有相同类型状态所需的时间。