Abstract
We study the online maximum coverage problem on a line, in which, given an online sequence of sub-intervals (which may intersect among each other) of a target large interval and an integer k, we aim to select at most k of the sub-intervals such that the total covered length of the target interval is maximized. The decision to accept or reject each sub-interval is made immediately and irrevocably (no preemption) right at the release timestamp of the sub-interval. We comprehensively study different settings of the problem, regarding the number of total released sub-intervals, we consider the unique-number (UN) setting where the total number is known in advance and the arbitrary-number (AN) setting where the total number is not known, respectively; regarding the length of a released sub-interval, we generally consider three settings: each sub-interval is of a normalized unit-length (UL), a flexible-length (FL) in a known range, or an arbitrary-length (AL). In addition, we extend the UL setting to a generalized unit-sum (US) setting, where a batch of a finite number of disjoint sub-intervals of the unit total length is released instead at each timestamp, and accordingly k batches can be accepted. We first prove in the AL setting that no online deterministic algorithm can achieve a bounded competitive ratio. Then, we present lower bounds on the competitive ratio for the other settings concerned in this paper. For the offline problem where the sequence of all the released sub-intervals is known in advance to the decision-maker, we propose a dynamic-programming-based optimal approach as the benchmark. For the online problem, we first propose a single-threshold-based deterministic algorithm SOA by adding a sub-interval if the added length exceeds a certain threshold, achieving competitive ratios close to the lower bounds, respectively. Then, we extend to a double-thresholds-based algorithm DOA, by using the first threshold for exploration and the second threshold (larger than the first one) for exploitation. With the two thresholds solved by our proposed program, we show that DOA improves SOA in the worst-case performance. Moreover, we prove that a deterministic algorithm that accepts sub-intervals by multi non-increasing thresholds cannot outperform even SOA.
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Notes
- 1.
When \(k=1\), our problem degenerates to the classical secretary problem without expertise sub-interval overlap.
- 2.
We do not distinguish our offline solution in the other dimension since our solution performs optimally in either the UN or the AN.
- 3.
The major difference between \(\kappa (\mathbb {V}_i,j)\) and \(\chi (\mathbb {V}_i,j)\) is that \(\kappa (\mathbb {V}_i,j)\) always accepts the last sub-interval \(V_i\) in \(\mathbb {V}_i\) while \(\chi (\mathbb {V}_i,j)\) does not necessarily.
- 4.
Constraint (vii) actually can be restricted, by calculation, to \(\left\lceil \frac{k+1}{5} \right\rceil \le \omega \le k\).
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Acknowledgements
This work was done when Songhua Li was visiting the Singapore University of Technology and Design. Minming Li is also from City University of Hong Kong Shenzhen Research Institute, Shenzhen, P.R. China. The work described in this paper was partially supported by Project 11771365 supported by NSFC. We would like to thank all the reviewers for their comments.
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Li, S., Li, M., Duan, L., Lee, V.C.S. (2020). Online Maximum k-Interval Coverage Problem. In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_31
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