  The Moran process is a classic stochastic process that models the rise and
takeover of novel traits in network-structured populations. In biological
terms, a set of mutants, each with fitness $m\in(0,\infty)$ invade a population
of residents with fitness $1$. Each agent reproduces at a rate proportional to
its fitness and each offspring replaces a random network neighbor. The process
ends when the mutants either fixate (take over the whole population) or go
extinct. The fixation probability measures the success of the invasion. To
account for environmental heterogeneity, we study a generalization of the
Standard process, called the Heterogeneous Moran process. Here, the fitness of
each agent is determined both by its type (resident/mutant) and the node it
occupies. We study the natural optimization problem of seed selection: given a
budget $k$, which $k$ agents should initiate the mutant invasion to maximize
the fixation probability? We show that the problem is strongly inapproximable:
it is $\mathbf{NP}$-hard to distinguish between maximum fixation probability 0
and 1. We then focus on mutant-biased networks, where each node exhibits at
least as large mutant fitness as resident fitness. We show that the problem
remains $\mathbf{NP}$-hard, but the fixation probability becomes submodular,
and thus the optimization problem admits a greedy $(1-1/e)$-approximation. An
experimental evaluation of the greedy algorithm along with various heuristics
on real-world data sets corroborates our results.
