Interval Selection with Binary Predictions

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. 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.