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An \(O(\lg \lg {\mathrm {OPT}})\)-Approximation Algorithm for Multi-guarding Galleries

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Abstract

We consider a generalization of the familiar art gallery problem in which individual points within the gallery need to be visible to some specified, but not necessarily uniform, number of guards. We provide an \(O(\lg \lg {\mathrm {OPT}})\)-approximation algorithm for this multi-guarding problem in simply-connected polygonal regions, with a minimum number (\({\mathrm {OPT}}\)) of vertex guards (possibly co-located). Our approximation algorithm is based on a polynomial-time algorithm for building what we call \(\varepsilon \)-multinets of size \(O\left( \frac{1}{\varepsilon }\lg \lg \frac{1}{\varepsilon }\right) \) for the instances of Multi-HittingSet  associated with our multi-guarding problem. We then apply a now-standard linear-programming technique to build an approximation algorithm from this \(\varepsilon \)-multinet finder. This paper corrects, and simplifies the analysis of, the \(O\left( \frac{1}{\varepsilon }\lg \lg \frac{1}{\varepsilon }\right) \)-time \(\varepsilon \)-net-finder described in [26], that was used to build an \(O(\lg \lg {\mathrm {OPT}})\)-approximation algorithm for the standard guarding problem in which all points within the gallery are required to be visible to at least one guard.

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Notes

  1. See Appendix for a comparison of our approach with that of [26], including a discussion of the source of the difficulty with the latter.

  2. It is a pseudopolynomial-time algorithm in that its running time may be linear in the ratio between the longest and shortest distances between two vertices.

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Acknowledgments

The author acknowledges the generous support of the Natural Sciences and Engineering Research Council of Canada. The many helpful discussions with Daniel Busto and William Evans concerning multi-guarding problems are also gratefully acknowledged.

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  1. Department of Computer Science, University of British Columbia, Vancouver, BC, Canada

    David Kirkpatrick

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Appendix: Remark on the Hierarchical Fragmentation Construction of King and Kirkpatrick [26]

Appendix: Remark on the Hierarchical Fragmentation Construction of King and Kirkpatrick [26]

A very similar hierarchical fragmentation (differing from ours only in the definition of \(t\), and in the level-1 fragmentation factor \(b_1\)) was described by King and Kirkpatrick [26] in developing their approximation bound for optimal \(1\)-guarding. Unfortunately, the choice of \(\alpha \) (which, together with \(t\) determines \(b_1\)) given in their Eq. (3) does not always guarantee that their Eq. (1) holds. In particular, consider the case when \(1/ \varepsilon = 2^{2^{t-1}+1}\) (so \(t = \lceil \log \log (1/ \varepsilon ) \rceil \), as specified). In this case, \(\alpha = 1/ (4t 2^{2^{t-1}+1-t})\) and so \(t \alpha 2^{2^t}\) (the bound on \(|S_{HF}|\), the size of their guard set) is essentially \(2^t \cdot 1/ \varepsilon \), which is \(\Theta ( (1/ \varepsilon ) \log (1/ \varepsilon ))\), not \(O((1/ \varepsilon ) \log \log (1/ \varepsilon ))\), as claimed in their Eq. (1).

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Kirkpatrick, D. An \(O(\lg \lg {\mathrm {OPT}})\)-Approximation Algorithm for Multi-guarding Galleries. Discrete Comput Geom 53, 327–343 (2015). https://doi.org/10.1007/s00454-014-9656-8

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