  Interval-constrained online bipartite matching problem frequently occurs in
medical appointment scheduling: unit-time jobs representing patients arrive
online and are assigned to a time slot within their given time interval. We
consider a variant of this problem where reassignments are allowed and extend
it by a notion of current time, which is decoupled from the job arrival events.
As jobs appear, the current point in time gradually advances. Jobs that are
assigned to the current time unit become processed, which fixes part of the
matching and disables these jobs or slots for reassignments in future steps. We
refer to these time-dependent restrictions on reassignments as the over-time
property.
  We show that FirstFit with reassignments according to the shortest augmenting
path rule is $\frac{2}{3}$-competitive with respect to the matching
cardinality, and that the bound is tight. Interestingly, this bound holds even
if the number of reassignments per job is bound by a constant. For the number
of reassignments performed by the algorithm, we show that it is in $\Omega(n
\log n)$ in the worst case, where $n$ is the number of patients or jobs on the
online side. This result is in line with lower bounds for the number of
reassignments in online bipartite matching with reassignments, and, similarly
to this previous work, we also conjecture that this bound should be tight.
Known upper bounds like the $O(n \log^2 n)$ for online bipartite matching with
reassignments by Bernstein, Holm, and Rotenberg do not transfer directly: while
our interval constraints simplify the problem, the over-time property restricts
the set of possible reassignments.
