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 $$\frac{2}{3}$$ 2 3 -competitive with respect to the matching cardinality. The competitive ratio remains $$\frac{2}{3}$$ 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 $$\frac{1}{2}$$ 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 $$\Omega (n^2)$$ Ω ( 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 $$\frac{2}{3}$$ 2 3 -competitive, which is optimal with respect to the matching cardinality, while EDF loses its optimality.

In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, we decide for each step how long we select the value. Then we cannot select another value until this period is over. The goal is to maximize the expectation of the sum of selected values. We describe the structure of the optimal stopping rule and give upper and lower bounds on the prophet inequality. In online algorithms terminology, this corresponds to bounds on the competitive ratio of an online algorithm.

We study packing linear programs (LPs) in an online model where the columns are presented to the algorithm in random order. This natural problem was investigated in various recent studies motivated, e.g., by online ad allocations and yield management, where rows correspond to resources and columns to requests specifying demands for resources. Our main contribution is a $1-O(\sqrt{\nicefrac{(\log d)}{B}})$-competitive online algorithm. Here $d$ denotes the column sparsity, i.e., the maximum number of resources that occur in a single column, and $B$ denotes the capacity ratio $B$, i.e., the ratio between the capacity of a resource and the maximum demand for this resource. In other words, we achieve a $(1-\epsilon)$-approximation if the capacity ratio satisfies $B=\Omega(\frac{\log d}{\epsilon^2})$, which is known to be the best possible for any (randomized) online algorithms. Our result improves exponentially on previous work with respect to the capacity ratio. In contrast to existing results on packing LP problems, our algorithm does not use dual prices to guide the allocation of resources over time. Instead, the algorithm simply solves, for each request, a scaled version of the partially known primal program and randomly rounds the obtained fractional solution to obtain an integral allocation for this request. We show that this simple algorithmic technique is not restricted to packing LPs with large capacity ratio of order $\Omega(\log d)$, but also yields close-to-optimal competitive ratios if the capacity ratio is bounded by a constant. In particular, we prove an upper bound on the competitive ratio of $\Omega(d^{\nicefrac{-1}{(B-1)}})$ for any $B\geq2$. In addition, we show that our approach can be combined with VCG payments and obtain an incentive-compatible $(1-\epsilon)$-competitive mechanism for packing LPs with $B=\Omega(\frac{\log m}{\epsilon^2})$, where $m$ is the number of constraints. Finally, we apply our technique to the generalized assignment problem for which we obtain the first online algorithm with competitive ratio $O(1)$.

This paper considers optimizing a submodular function subject to a set of downward closed constraints. Previous literature on this problem has often constructed solutions by (1) discovering a fractional solution to the multi-linear extension and (2) rounding this solution to an integral solution via a contention resolution scheme. This line of research has improved results by either optimizing (1) or (2). Diverging from previous work, this paper introduces a principled method called contention resolution extensions of submodular functions. A contention resolution extension combines the contention resolution scheme into a continuous extension of a discrete submodular function. The contention resolution extension can be defined from effectively any contention resolution scheme. In the case where there is a loss in both (1) and (2), by optimizing them together, the losses can be combined resulting in an overall improvement. This paper showcases the concept by demonstrating that for the problem of optimizing a non-monotone submodular subject to the elements forming an independent set in an interval graph, the algorithm gives a .188-approximation. This improves upon the best known 1/(2e)~eq .1839 approximation.

The Temp Secretary Problem was recently introduced by [Fiat et al., ESA 2015]. It is a generalization of the Secretary Problem, in which commitments are temporary for a fixed duration. We present a simple online algorithm with improved performance guarantees for cases already considered by [Fiat et al., ESA 2015] and give competitive ratios for new generalizations of the problem. In the classical setting, where candidates have identical contract durations gamma << 1 and we are allowed to hire up to B candidates simultaneously, our algorithm is (1/2) - O(sqrt{gamma})-competitive. For large B, the bound improves to 1 - O(1/sqrt{B}) - O(sqrt{gamma}). Furthermore we generalize the problem from cardinality constraints towards general packing constraints. We achieve a competitive ratio of 1 - O(sqrt{(1+log(d) + log(B))/B}) - O(sqrt{gamma}), where d is the sparsity of the constraint matrix and B is generalized to the capacity ratio of linear constraints. Additionally we extend the problem towards arbitrary hiring durations. Our algorithmic approach is a relaxation that aggregates all temporal constraints into a non-temporal constraint. Then we apply a linear scaling algorithm that, on every arrival, computes a tentative solution on the input that is known up to this point. This tentative solution uses the non-temporal, relaxed constraints scaled down linearly by the amount of time that has already passed.

We study packing LPs in an online model where the columns are presented to the algorithm in random order. This natural problem was investigated in various recent studies motivated, e.g., by online ad allocations and yield management where rows correspond to resources and columns to requests specifying demands for resources. Our main contribution is a 1 -- O(√(log d/B))-competitive online algorithm. Here d denotes the column sparsity, i.e., the maximum number of resources that occur in a single column, and B denotes the capacity ratio B, i.e., the ratio between the capacity of a resource and the maximum demand for this resource. In other words, we achieve a (1--ε)-approximation if the capacity ratio satisfies B=Ω(logd/ε2), which is known to be best-possible for any (randomized) online algorithms.