Interval-constrained bipartite matching over time

Abstract

In medical appointment assignment, unit jobs representing patients arrive online and are assigned to a time slot within their given feasible time interval. We model this setting as interval-constrained online bipartite matching problem. 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 arrive, 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 this problem with respect to the resulting matching size and the number of occurring reassignments. We show that FirstFit with reassignments according to the shortest augmenting path rule is exactly \(\frac{2}{3}\)-competitive with respect to the matching cardinality. The competitive ratio remains \(\frac{2}{3}\) if we restrict FirstFit to consider only augmenting paths causing at most a constant number of reassignments, which implies a linear number of reassignments in total. This fills the gap between the known optimal algorithm with no reassignments at all, which is \(\frac{1}{2}\)-competitive, on the one hand, and an earliest-deadline-first strategy (EDF), which we prove to be 1-competitive in our over-time framework, but which suffers \(\Omega (n^2)\) reassignments in the worst case, on the other. We further extend the problem setting to the sets of feasible slots per job that are not intervals. In this setting, FirstFit remains \(\frac{2}{3}\)-competitive, which is optimal with respect to the matching cardinality, while EDF loses its optimality.