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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.

Aritra Banik, Rajiv Raman, and Saurabh Ray. On Geometric Priority Set Cover Problems. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{banik_et_al:LIPIcs.ISAAC.2021.12, author = {Banik, Aritra and Raman, Rajiv and Ray, Saurabh}, title = {{On Geometric Priority Set Cover Problems}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.12}, URN = {urn:nbn:de:0030-drops-154459}, doi = {10.4230/LIPIcs.ISAAC.2021.12}, annote = {Keywords: Approximation algorithms, geometric set cover, local search, quasi-uniform sampling} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

In the Set Multicover problem, we are given a set system (X,𝒮), where X is a finite ground set, and 𝒮 is a collection of subsets of X. Each element x ∈ X has a non-negative demand d(x). The goal is to pick a smallest cardinality sub-collection 𝒮' of 𝒮 such that each point is covered by at least d(x) sets from 𝒮'. In this paper, we study the set multicover problem for set systems defined by points and non-piercing regions in the plane, which includes disks, pseudodisks, k-admissible regions, squares, unit height rectangles, homothets of convex sets, upward paths on a tree, etc.
We give a polynomial time (2+ε)-approximation algorithm for the set multicover problem (P, ℛ), where P is a set of points with demands, and ℛ is a set of non-piercing regions, as well as for the set multicover problem (𝒟, P), where 𝒟 is a set of pseudodisks with demands, and P is a set of points in the plane, which is the hitting set problem with demands.

Rajiv Raman and Saurabh Ray. Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{raman_et_al:LIPIcs.ESA.2020.78, author = {Raman, Rajiv and Ray, Saurabh}, title = {{Improved Approximation Algorithm for Set Multicover with Non-Piercing Regions}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {78:1--78:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.78}, URN = {urn:nbn:de:0030-drops-129441}, doi = {10.4230/LIPIcs.ESA.2020.78}, annote = {Keywords: Approximation algorithms, geometry, Covering} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Let C be a bounded convex object in R^d, and P a set of n points lying outside C. Further let c_p, c_q be two integers with 1 <= c_q <= c_p <= n - floor[d/2], such that every c_p + floor[d/2] points of P contains a subset of size c_q + floor[d/2] whose convex-hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex-hull is disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.
In particular, our general theorem implies polynomial bounds for Hadwiger-Debrunner (p, q) numbers for balls in R^d. For example, it follows from our theorem that when p > q >= (1+beta) * d/2 for beta > 0, then any set of balls satisfying the HD(p,q) property can be hit by O(q^2 p^{1+1/(beta)} log p) points. This is the first improvement over a nearly 60-year old exponential bound of roughly O(2^d).
Our results also complement the results obtained in a recent work of Keller et al. where, apart from improvements to the bound on HD(p, q) for convex sets in R^d for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

Nabil H. Mustafa and Saurabh Ray. On a Problem of Danzer. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 64:1-64:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{mustafa_et_al:LIPIcs.ESA.2018.64, author = {Mustafa, Nabil H. and Ray, Saurabh}, title = {{On a Problem of Danzer}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {64:1--64:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.64}, URN = {urn:nbn:de:0030-drops-95271}, doi = {10.4230/LIPIcs.ESA.2018.64}, annote = {Keywords: Convex polytopes, Hadwiger-Debrunner numbers, Epsilon-nets, Balls} }

Document

**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Given a hypergraph H=(X,S), a planar support for H is a planar graph G with vertex set X, such that for each hyperedge S in S, the sub-graph of G induced by the vertices in S is connected. Planar supports for hypergraphs have found several algorithmic applications, including several packing and covering problems, hypergraph coloring, and in hypergraph visualization.
The main result proved in this paper is the following: given two families of regions R and B in the plane, each of which consists of connected, non-piercing regions, the intersection hypergraph H_R(B) = (B, {B_r}_{r in R}), where B_r = {b in B: b cap r != empty set} has a planar support. Further, such a planar support can be computed in time polynomial in |R|, |B|, and the number of vertices in the arrangement of the regions in R cup B. Special cases of this result include the setting where either the family R, or the family B is a set of points.
Our result unifies and generalizes several previous results on planar supports, PTASs for packing and covering problems on non-piercing regions in the plane and coloring of intersection hypergraph of non-piercing regions.

Rajiv Raman and Saurabh Ray. Planar Support for Non-piercing Regions and Applications. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 69:1-69:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{raman_et_al:LIPIcs.ESA.2018.69, author = {Raman, Rajiv and Ray, Saurabh}, title = {{Planar Support for Non-piercing Regions and Applications}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {69:1--69:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.69}, URN = {urn:nbn:de:0030-drops-95320}, doi = {10.4230/LIPIcs.ESA.2018.69}, annote = {Keywords: Geometric optimization, packing and covering, non-piercing regions} }

Document

**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). We show that the local
search algorithm that yields PTASs when the regions are disks [Aschner/Katz/Morgenstern/Yuditsky, WALCOM 2013; Gibson/Pirwani, 2005; Mustafa/Raman/Ray, 2015] can be extended to work for non-piercing regions. While such an extension is intuitive and natural, attempts to settle this question have failed even for pseudodisks. The techniques used for analysis when the regions are disks rely heavily on the underlying geometry, and do not extend to topologically defined settings such as pseudodisks. In order to prove our results, we introduce novel techniques that we believe will find applications in other problems.
We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded. Our result settles a conjecture of Har-Peled (see Conclusion of [Har-Peled, SoCG 2014]) in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane.
Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our
objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity, extending one of the results of Ene et al. [Ene/Har-Peled/Raichel, SoCG 2012].

Sathish Govindarajan, Rajiv Raman, Saurabh Ray, and Aniket Basu Roy. Packing and Covering with Non-Piercing Regions. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{govindarajan_et_al:LIPIcs.ESA.2016.47, author = {Govindarajan, Sathish and Raman, Rajiv and Ray, Saurabh and Basu Roy, Aniket}, title = {{Packing and Covering with Non-Piercing Regions}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {47:1--47:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.47}, URN = {urn:nbn:de:0030-drops-63591}, doi = {10.4230/LIPIcs.ESA.2016.47}, annote = {Keywords: Local Search, Set Cover, Dominating Set, Capacitated Packing, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others).
Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding - both combinatorial and algorithmic - of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3,2)-local search and give an (8+\epsilon)-approximation algorithm with expected running time ˜O(n^{2.34}); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n^{15}) -- that too just for unit disks. The techniques and ideas generalize to (4,3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3,2) local search gives an 8-approximation and no better \footnote{This is assuming the use of the standard framework. Improvement of the approximation factor by using additional properties specific to the problem may be possible.}. Similarly (4,3)-local search gives a 5-approximation for all these problems.

Norbert Bus, Shashwat Garg, Nabil H. Mustafa, and Saurabh Ray. Improved Local Search for Geometric Hitting Set. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 184-196, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bus_et_al:LIPIcs.STACS.2015.184, author = {Bus, Norbert and Garg, Shashwat and Mustafa, Nabil H. and Ray, Saurabh}, title = {{Improved Local Search for Geometric Hitting Set}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {184--196}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.184}, URN = {urn:nbn:de:0030-drops-49135}, doi = {10.4230/LIPIcs.STACS.2015.184}, annote = {Keywords: hitting sets, Delaunay triangulation, local search, disks, geometric algorithms} }

Document

**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

The existence of Macbeath regions is a classical theorem in convex geometry ("A Theorem on non-homogeneous lattices", Annals of Math, 1952). We refer the reader to the survey of I. Barany for several applications. Recently there have been some striking applications of Macbeath regions in discrete and computational geometry.
In this paper, we study Macbeath's problem in a more general setting, and not only for the Lebesgue measure as is the case in the classical theorem. We prove near-optimal generalizations for several basic geometric set systems. The problems and techniques used are closely linked to the study of espilon-nets for geometric set systems.

Nabil H. Mustafa and Saurabh Ray. Near-Optimal Generalisations of a Theorem of Macbeath. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 578-589, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{mustafa_et_al:LIPIcs.STACS.2014.578, author = {Mustafa, Nabil H. and Ray, Saurabh}, title = {{Near-Optimal Generalisations of a Theorem of Macbeath}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {578--589}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.578}, URN = {urn:nbn:de:0030-drops-44890}, doi = {10.4230/LIPIcs.STACS.2014.578}, annote = {Keywords: Epsilon Nets, Cuttings, Union Complexity, Geometric Set systems, Convex Geometry} }

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