  We address the problem of finding sets of integers of a given size with
maximum number of pairs summing to powers of $2$. By fixing particular pairs
this problem reduces to finding a labeling of the vertices of a given graph
with pairwise distinct integers such that the endpoint labels for each edge sum
to a power of $2$. We propose an efficient algorithm for this problem, which we
use to determine the maximum size of graphs of order $n$ that admit such a
labeling for all $n\leq 18$. We also identify the minimal forbidden subgraphs
of order $\leq 11$, whose presence prevents graphs from having such a labeling.
