Following a line of work that takes advantage of vast machine-learned data to enhance online algorithms with (possibly erroneous) information about future inputs, we consider predictions in the context of deterministic algorithms for the problem of selecting a maximum weight independent set of intervals arriving on the real line. We look at two weight functions, unit (constant) weights, and weights proportional to the interval’s length. In the classical online model of irrevocable decisions, no algorithm can achieve constant competitiveness. In this setting, we show that a simple algorithm that is faithful to the predictions is optimal, and achieves an objective value of at least OPT - η, with η being the total error in the predictions, both for unit, and proportional weights. When revocable acceptances (a form of preemption) are allowed, the optimal deterministic algorithm for unit weights is 2k-competitive, where k is the number of different interval lengths. We give an algorithm with performance OPT − η (and therefore 1-consistent), that is also (2k + 1)-robust. For proportional weights, there is an optimal (2φ + 1)-competitive algorithm, where φ is the golden ratio. We present an algorithm with parameter λ > 1 that is 3λ / (λ - 1) -consistent, and (4λ^2 + 2λ) / (λ - 1)-robust. Although these bounds are not tight, we show that for λ > 3.42 we achieve consistency better than the optimal online guarantee, while maintaining bounded robustness. We conclude with some experimental results on real-world data that complement our theoretical findings, and show the benefit of prediction algorithms for online interval selection, even in the presence of high error.

We introduce a new random input model for bipartite matching which we call the Random Type Poisson Arrival Model. Just like in the known i.i.d. model (introduced by Feldman et al. [Feldman et al., 2009]), online nodes have types in our model. In contrast to the adversarial types studied in the known i.i.d. model, following the random graphs studied in Mastin and Jaillet [A. Mastin, 2013], in our model each type graph is generated randomly by including each offline node in the neighborhood of an online node with probability c/n independently. In our model, nodes of the same type appear consecutively in the input and the number of times each type node appears is distributed according to the Poisson distribution with parameter 1. We analyze the performance of the simple greedy algorithm under this input model. The performance is controlled by the parameter c and we are able to exactly characterize the competitive ratio for the regimes c = o(1) and c = omega(1). We also provide a precise bound on the expected size of the matching in the remaining regime of constant c. We compare our results to the previous work of Mastin and Jaillet who analyzed the simple greedy algorithm in the G_{n,n,p} model where each online node type occurs exactly once. We essentially show that the approach of Mastin and Jaillet can be extended to work for the Random Type Poisson Arrival Model, although several nontrivial technical challenges need to be overcome. Intuitively, one can view the Random Type Poisson Arrival Model as the G_{n,n,p} model with less randomness; that is, instead of each online node having a new type, each online node has a chance of repeating the previous type.