  For a fixed integer r, consider the following random process. At each round,
one is presented with r random edges from the edge set of the complete graph on
n vertices, and is asked to choose one of them. The selected edges are
collected into a graph, which thus grows at the rate of one edge per round.
This is a natural generalization of what is known in the literature as an
Achlioptas process (the original version has r=2), which has been studied by
many researchers, mainly in the context of delaying or accelerating the
appearance of the giant component.
  In this paper, we investigate the small subgraph problem for Achlioptas
processes. That is, given a fixed graph H, we study whether there is an online
algorithm that substantially delays or accelerates a typical appearance of H,
compared to its threshold of appearance in the random graph G(n, M). It is easy
to see that one cannot accelerate the appearance of any fixed graph by more
than the constant factor r, so we concentrate on the task of avoiding H. We
determine thresholds for the avoidance of all cycles C_t, cliques K_t, and
complete bipartite graphs K_{t,t}, in every Achlioptas process with parameter r
>= 2.
