Interval Selection with Binary Predictions
Christodoulos Karavasilis
TL;DR
The paper studies online interval selection with binary predictions for unit and proportional weights under irrevocable and revocable acceptance models. It proves that a predictions-faithful strategy is optimal in the irrevocable setting, achieving $ALG \ge OPT - \eta$ for both weight regimes. In the revocable setting, it introduces Revoke-Unit and Revoke-Proportional, delivering 1-consistency with predictable robustness bounds that depend on the number of interval lengths or the prediction parameter $\lambda$, respectively, and shows how larger trust in predictions can improve consistency while keeping robustness bounded. The theoretical results are complemented by experiments on real scheduling datasets, demonstrating practical gains from using imperfect predictions even under significant error. Overall, the work narrows the gap between online performance and offline benchmarks by leveraging succinct predictions in online interval selection.
Abstract
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 (Bachmann et al. [BHS13] for unit, Lipton and Tomkins [LT94] for proportional). 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 [BK23], 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, Garay et al. [GGKMY97] give an optimal $(2φ+ 1)$-competitive algorithm, where $φ$ is the golden ratio. We present an algorithm with parameter $λ> 1$ that is $\frac{3λ}{λ-1}$-consistent, and $\frac{4λ^2 +2λ}{λ-1}$-robust. Although these bounds are not tight, we show that for $λ> 3.42$ we achieve consistency better than the optimal online guarantee in [GGKMY97], 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.