  To gain a comprehensive view of what the LHC tells us about physics beyond
the Standard Model (BSM), it is crucial that different BSM-sensitive analyses
can be combined. But in general, search analyses are not statistically
orthogonal, so performing comprehensive combinations requires knowledge of the
extent to which the same events co-populate multiple analyses' signal regions.
We present a novel, stochastic method to determine this degree of overlap and a
graph algorithm to efficiently find the combination of signal regions with no
mutual overlap that optimises expected upper limits on BSM-model
cross-sections. The gain in exclusion power relative to single-analysis limits
is demonstrated with models with varying degrees of complexity, ranging from
simplified models to a 19-dimensional supersymmetric model.
