13 research outputs found
Packing and Covering with Non-Piercing Regions
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]
Discrete Helly-type theorems for pseudohalfplanes
We prove discrete Helly-type theorems for pseudohalfplanes, which extend
recent results of Jensen, Joshi and Ray about halfplanes. Among others we show
that given a family of pseudohalfplanes and a set of points , if
every triple of pseudohalfplanes has a common point in then there exists a
set of at most two points that hits every pseudohalfplane of . We also
prove that if every triple of points of is contained in a pseudohalfplane
of then there are two pseudohalfplanes of that cover all
points of .
To prove our results we regard pseudohalfplane hypergraphs, define their
extremal vertices and show that these behave in many ways as points on the
boundary of the convex hull of a set of points. Our methods are purely
combinatorial
Covering Problems via Structural Approaches
The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible.
In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide:
- An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC.
- A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure.
- A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity.
- Applications of the above to various capacitated covering problems via linear programming strengthening and rounding.
In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability
On Geometric Priority Set Cover Problems
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
More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time
We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D . More precisely, we obtain randomized update time O(n^{2/3+?}) for an arbitrarily small constant ? > 0. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D . As a byproduct, our techniques for dynamic set cover also yield an optimal randomized O(nlog n)-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier O(nlog n(log log n)^{O(1)}) result [SoCG 2020]
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Enclosing Points with Geometric Objects
Let be a set of points in and be a set of
geometric objects in , where . We study
the problem of computing a minimum subset
that encloses all points in . Here a point is enclosed by
if it lies in a bounded connected component of . We propose two algorithmic
frameworks to design polynomial-time approximation algorithms for the problem.
The first framework is based on sparsification and min-cut, which results in
-approximation algorithms for unit disks, unit squares, etc. The second
framework is based on LP rounding, which results in an -approximation algorithm for segments, where is the inverse
Ackermann function, and an -approximation algorithm for disks.Comment: In SoCG'2
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum