Abstract
We study the online maximum coverage problem on a target interval, in which, given an online sequence of sub-intervals (which may intersect among each other) to arrive, 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 right at the release time of the sub-interval. We comprehensively study various settings of this problem regarding both the length of each released sub-interval and the total number of released sub-intervals. To begin with, we investigate the offline version of the problem where the sequence of all the released sub-intervals is known in advance to the decision-maker and propose two polynomial-time optimal solutions to different settings of our offline problem. For the online problem, lower bounds on the competitive ratio are first proposed on our well-designed release schemes of sub-intervals. Then, we propose a Single-threshOld-based deterministic Algorithm (SOA), which adds a sub-interval if the added length without overlap exceeds a certain threshold, achieving competitive ratios close to the lower bounds. Further, we extend SOA to a Double-threshOlds-based deterministic Algorithm (DOA) by using the first threshold for exploration and the second threshold (larger than the first one) for exploitation. With the two thresholds generated by our proposed program, we show that DOA outperforms SOA slightly in the worst-case scenario. Moreover, we show that more thresholds cannot induce better worst-case performance of an online deterministic algorithm as long as those thresholds are used in non-increasing order in accepting sub-intervals.
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Notes
This target interval represents the expertise distribution of each online worker, which is in the simplest but fundamental way. For example, a worker’s expertise distribution is close to the left boundary of the target interval if her expertise is in combinatorial optimization, while another with expertise distributed close to the right boundary implies a game theory background.
When \(k=1\), our problem degenerates to the classical secretary problem without expertise sub-interval overlap.
The length of each sub-interval is normalized with regard to a.
This models the scenario where an online worker masters different domains of expertise. Moreover, we keep the same unit-sum for all the sub-intervals to show similar strength of all the online workers.
We do not distinguish our offline solution between FN and VN settings since our solution performs optimally in each of them.
This implies that OPT remains quota j, which is not sufficient to accept all the remaining sub-intervals in \({\mathbb {V}}_i\). In addition, we have \(i\ge 2\) in this case.
Note that 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.
Our approach to counting the contribution of each accepted sub-interval guarantees that the length contribution of sub-intervals accepted in the future will be independent of any accepted sub-interval, which provides convenience in formulating the \( \chi ({\mathbb {V}}_i,j)\) in Case 2.
Note that \(i=0\) would never happen, which is ensured by Case 2 and the basic assumption \(n\ge k+1\). We include the condition \(i=0\) here for the sake of completeness.
Note that Klee’s algorithm (Klee 1977) can compute the length of an union of x sub-intervals in \(O(x\log x)\) time, which is based on sorting the intervals. Accordingly, one can compute the length of x sorted sub-intervals in O(x) time by scanning the x sub-intervals.
Since an iteration will stop when \(|{\mathbb {V}}_i|\le j\) (see Case 2 of our offline solution). The overall states of our solution is improved to \(O(k(n-k))\).
When some \(V_q\) contains less than u of small sub-intervals, one can copy a small sub-interval of \(V_q\) several times until the total number of small sub-intervals of \(V_q\) reaches u, which does not affect the length contribution of \(V_q\) to the solution.
Here, the “best-fit \(\theta \)" refers to the \(\theta \) that minimizes the upper bound as presented in Theorem 4.
Note that the length of each of these \(\mu \) disjoint intervals could be larger than one as an interval could be formed by multiple accepted sub-intervals.
Constraint (vii) actually can be restricted, by calculation, to \(\left\lceil \frac{k+1}{5} \right\rceil \le \omega \le k\).
Here, the j is chosen such that \(2j+2\le k<3j\).
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Acknowledgements
We thank anonymous reviewers for their valuable comments. A preliminary version of this work appeared in the proceedings of COCOA 2020 (Li et al. 2020).
Funding
The research of Lingjie Duan and Songhua Li was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 2 Grant (Award No. MOE-T2EP20121-0001).
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Li, S., Li, M., Duan, L. et al. Online algorithms for the maximum k-interval coverage problem. J Comb Optim 44, 3364–3404 (2022). https://doi.org/10.1007/s10878-022-00898-3
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DOI: https://doi.org/10.1007/s10878-022-00898-3