  We analyse the asymptotic behaviour of random instances of the Maximum Set
Packing (MSP) optimization problem, also known as Maximum Matching or Maximum
Strong Independent Set on Hypergraphs. We give an analytical prediction of the
MSPs size using the 1RSB cavity method from statistical mechanics of disordered
systems. We also propose a heuristic algorithm, a generalization of the
celebrated Karp-Sipser one, which allows us to rigorously prove that the
replica symmetric cavity method prediction is exact for certain problem
ensembles and breaks down when a core survives the leaf removal process. The
$e$-phenomena threshold discovered by Karp and Sipser, marking the onset of
core emergence and of replica symmetry breaking, is elegantly generalized to
$c_s = \frac{e}{d-1}$ for one of the ensembles considered, where $d$ is the
size of the sets.
