Interval-Constrained Bipartite Matching over Time

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 feasible time interval. We consider a variant of this problem where reassignments are allowed and extend it by a notion of time that is decoupled from the job arrival events. As jobs appear, the current point in time gradually advances, and once the time of a slot is passed, the job assigned to it is fixed and cannot be reassigned anymore. We analyze two algorithms for the problem with respect to the resulting matching size and the number of reassignments they make. 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. For the number of reassignments performed by the algorithm, we show that it is in $Ω(n \log n)$ in the worst case, where $n$ is the number of patients or jobs on the online side. The competitive ratio remains bounded by $\frac{2}{3}$ if we restrict the algorithm to make only up to a constant number $k \geq 1$ of reassignments per job arrival. This fills the gap between the known optimal algorithm that makes no reassignments, which is $\frac{1}{2}$-competitive, on the one hand, and an earliest-deadline-first strategy (EDF), which we prove to obtain a maximum matching in this over-time framework, but which suffers $Ω(n^2)$ reassignments in the worst case, on the other hand. Further, we consider the setting in which the sets of feasible slots per job that are not intervals. We show that FirstFit remains $\frac{2}{3}$-competitive in this case, and that this is the best possible deterministic competitive ratio, while EDF loses its optimality.