Abstract
Given an interval I = {1,2, ..., n} of points, a collection \(\mathcal I\) of subintervals of I and a fraction 0 ≤ r ≤ 1, we consider the following variation of partial set cover. We wish to find an optimal subset of \(\mathcal I\) covering at least an r-fraction of I. While this problem is easily solved exactly in quadratic time using classical methods, we focus on developing scalable algorithms which return near-optimal solutions and run in near-linear time. We give a (1 + ε)-approximation algorithm running in \(O(\frac{1}{\epsilon} \cdot\min \{n + |\mathcal I|, |\mathcal I| \log |\mathcal I|\}\})\) time. We also prove a tight approximation ratio of 2 for a simple greedy algorithm for this problem, improving on the bound of 9 given in [10].
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Edwards, K., Griffiths, S., Kennedy, W.S. (2013). Partial Interval Set Cover – Trade-Offs between Scalability and Optimality. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_9
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