Minimum-Membership Geometric Set Cover, Revisited
We revisit a natural variant of the geometric set cover problem, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set S of points and a set ℛ of geometric objects, and the goal is to find a subset ℛ^* ⊆ ℛ to cover all points in S such that the membership of S with respect to ℛ^*, denoted by memb(S,ℛ^*), is minimized, where memb(S,ℛ^*) = max_{p ∈ S} |{R ∈ ℛ^*: p ∈ R}|. We give the first polynomial-time approximation algorithms for MMGSC in ℝ². Specifically, we achieve the following two main results.
- We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time n^{O(opt)} where opt is the optimum of the problem (i.e., the minimum membership).
- We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of o(log n) in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find ℛ^* ⊆ ℛ to cover S such that the ply of ℛ^* is minimized, where the ply is defined as the maximum number of objects in ℛ^* which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in O(n^{12}) time. We give a significantly simpler constant-approximation algorithm with near-linear running time.
geometric set cover
geometric optimization
approximation algorithms
Theory of computation~Design and analysis of algorithms
11:1-11:14
Regular Paper
https://arxiv.org/abs/2305.03985
The authors would like to thank Qizheng He, Daniel Lokshtanov, Rahul Saladi, Subhash Suri, and Haitao Wang for helpful discussions about the problems, and thank the anonymous reviewers for their detailed comments, which help significantly improve the writing of the paper.
Sayan
Bandyapadhyay
Sayan Bandyapadhyay
Portland State University, OR, USA
William
Lochet
William Lochet
LIRMM, Université de Montpellier, CNRS, Montpellier, France
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, Chennai, India
Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416), and Swarnajayanti Fellowship (No. DST/SJF/MSA01/2017-18).
Jie
Xue
Jie Xue
New York University Shanghai, China
https://orcid.org/0000-0001-7015-1988
10.4230/LIPIcs.SoCG.2023.11
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