  Inspired by real-world applications such as the assignment of pupils to
schools or the allocation of social housing, the one-sided matching problem
studies how a set of agents can be assigned to a set of objects when the agents
have preferences over the objects, but not vice versa. For fairness reasons,
most mechanisms use randomness, and therefore result in a probabilistic
assignment. We study the problem of decomposing these probabilistic assignments
into a weighted sum of ex-post (Pareto-)efficient matchings, while maximizing
the worst-case number of assigned agents. This decomposition preserves all the
assignments' desirable properties, most notably strategy-proofness. For a
specific class of probabilistic assignments, including the assignment by the
Probabilistic Serial mechanism, we propose a polynomial-time algorithm for this
problem that obtains a decomposition in which all matchings assign at least the
expected number of assigned agents by the probabilistic assignment, rounded
down, thus achieving the theoretically best possible guarantee. For general
probabilistic assignments, the problem becomes NP-hard. For the Random Serial
Dictatorship (RSD) mechanism, we show that the worst-case number of assigned
agents by RSD is at least half of the optimal, and that this bound is
asymptotically tight. Lastly, we propose a column generation framework for the
introduced problem, which we evaluate both on randomly generated data, and on
real-world school choice data from the Belgian cities Antwerp and Ghent.
