  We introduce the maximum $n$-times coverage problem that selects $k$ overlays
to maximize the summed coverage of weighted elements, where each element must
be covered at least $n$ times. We also define the min-cost $n$-times coverage
problem where the objective is to select the minimum set of overlays such that
the sum of the weights of elements that are covered at least $n$ times is at
least $\tau$. Maximum $n$-times coverage is a generalization of the multi-set
multi-cover problem, is NP-complete, and is not submodular. We introduce two
new practical solutions for $n$-times coverage based on integer linear
programming and sequential greedy optimization. We show that maximum $n$-times
coverage is a natural way to frame peptide vaccine design, and find that it
produces a pan-strain COVID-19 vaccine design that is superior to 29 other
published designs in predicted population coverage and the expected number of
peptides displayed by each individual's HLA molecules.
