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arXiv · 2110.14528

On Competitive Permutations for Set Cover by Intervals

Sariel Har-Peled·Jiaqi Cheng
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Abstract

We revisit the problem of computing an optimal partial cover of points by intervals. We show that the greedy algorithm computes a permutation $\Pi = \pi_1, \pi_2,\ldots$ of the intervals that is $3/4$-competitive for any prefix of $k$ intervals. That is, for any $k$, the intervals $\pi_1 \cup \cdots \cup \pi_k$ covers at least $3/4$-fraction of the points covered by the optimal solution using $k$ intervals. We also provide an approximation algorithm that, in $O(n + m/\varepsilon)$ time, computes a cover by $(1+\varepsilon)k$ intervals that is as good as the optimal solution using $k$ intervals, where $n$ is the number of input points, and $m$ is the number of intervals (we assume here the input is presorted). Finally, we show a counter example illustrating that the optimal solutions for set cover do not have the diminishing return property -- that is, the marginal benefit from using more sets is not monotonically decreasing. Fortunately, the diminishing returns does hold for intervals.

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Sariel Har-Peled, Jiaqi Cheng. 2021-10-27. On Competitive Permutations for Set Cover by Intervals. https://arxiv.org/abs/2110.14528

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