  Define Minimum Soapy Union (MinSU) as the following optimization problem:
given a $k$-tuple $(X_1, X_2,..., X_k)$ of finite integer sets, find a
$k$-tuple $(t_1, t_2,..., t_k)$ of integers that minimizes the cardinality of
$(X_1 + t_1) \cup (X_2 + t_2) \cup ... \cup (X_n + t_k)$. We show that MinSU is
NP-complete, APX-hard, and polynomial for fixed $k$. MinSU appears naturally in
the context of protein shotgun sequencing: Here, the protein is cleaved into
short and overlapping peptides, which are then analyzed by tandem mass
spectrometry. To improve the quality of such spectra, one then asks for the
mass of the unknown prefix (the shift) of the spectrum, such that the resulting
shifted spectra show a maximum agreement. For real-world data the problem is
even more complicated than our definition of MinSU; but our intractability
results clearly indicate that it is unlikely to find a polynomial time
algorithm for shotgun protein sequencing.
