6,910 research outputs found
A Constant Approximation for Colorful k-Center
In this paper, we consider the colorful k-center problem, which is a generalization of the well-known k-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius rho, such that with k balls of radius rho, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs
A Technique for Obtaining True Approximations for -Center with Covering Constraints
There has been a recent surge of interest in incorporating fairness aspects
into classical clustering problems. Two recently introduced variants of the
-Center problem in this spirit are Colorful -Center, introduced by
Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the
Fair Robust -Center problem introduced by Harris, Pensyl, Srinivasan, and
Trinh. To address fairness aspects, these models, compared to traditional
-Center, include additional covering constraints. Prior approximation
results for these models require to relax some of the normally hard
constraints, like the number of centers to be opened or the involved covering
constraints, and therefore, only obtain constant-factor pseudo-approximations.
In this paper, we introduce a new approach to deal with such covering
constraints that leads to (true) approximations, including a -approximation
for Colorful -Center with constantly many colors---settling an open question
raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan---and a
-approximation for Fair Robust -Center, for which the existence of a
(true) constant-factor approximation was also open. We complement our results
by showing that if one allows an unbounded number of colors, then Colorful
-Center admits no approximation algorithm with finite approximation
guarantee, assuming that . Moreover, under the
Exponential Time Hypothesis, the problem is inapproximable if the number of
colors grows faster than logarithmic in the size of the ground set
Techniques for Generalized Colorful k-Center Problems
Fair clustering enjoyed a surge of interest recently. One appealing way of integrating fairness aspects into classical clustering problems is by introducing multiple covering constraints. This is a natural generalization of the robust (or outlier) setting, which has been studied extensively and is amenable to a variety of classic algorithmic techniques. In contrast, for the case of multiple covering constraints (the so-called colorful setting), specialized techniques have only been developed recently for k-Center clustering variants, which is also the focus of this paper.
While prior techniques assume covering constraints on the clients, they do not address additional constraints on the facilities, which has been extensively studied in non-colorful settings. In this paper, we present a quite versatile framework to deal with various constraints on the facilities in the colorful setting, by combining ideas from the iterative greedy procedure for Colorful k-Center by Inamdar and Varadarajan with new ingredients. To exemplify our framework, we show how it leads, for a constant number ? of colors, to the first constant-factor approximations for both Colorful Matroid Supplier with respect to a linear matroid and Colorful Knapsack Supplier. In both cases, we readily get an O(2^?)-approximation.
Moreover, for Colorful Knapsack Supplier, we show that it is possible to obtain constant approximation guarantees that are independent of the number of colors ?, as long as ? = O(1), which is needed to obtain a polynomial running time. More precisely, we obtain a 7-approximation by extending a technique recently introduced by Jia, Sheth, and Svensson for Colorful k-Center
Non-Uniform k-Center and Greedy Clustering
In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k_i balls of radius r_i, for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded, assuming ≠ NP. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture - currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of k-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.publishedVersio
Quantitative combinatorial geometry for continuous parameters
We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems
where the sets involved are measured according to continuous functions such as
the volume or diameter. Among our results, we present continuous quantitative
versions of Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful
Carath\'eodory's theorem, and the colorful Tverberg theorem.Comment: 22 pages. arXiv admin note: substantial text overlap with
arXiv:1503.0611
Approximating Tverberg Points in Linear Time for Any Fixed Dimension
Let P be a d-dimensional n-point set. A Tverberg-partition of P is a
partition of P into r sets P_1, ..., P_r such that the convex hulls conv(P_1),
..., conv(P_r) have non-empty intersection. A point in the intersection of the
conv(P_i)'s is called a Tverberg point of depth r for P. A classic result by
Tverberg implies that there always exists a Tverberg partition of size n/(d+1),
but it is not known how to find such a partition in polynomial time. Therefore,
approximate solutions are of interest.
We describe a deterministic algorithm that finds a Tverberg partition of size
n/4(d+1)^3 in time d^{O(log d)} n. This means that for every fixed dimension we
can compute an approximate Tverberg point (and hence also an approximate
centerpoint) in linear time. Our algorithm is obtained by combining a novel
lifting approach with a recent result by Miller and Sheehy (2010).Comment: 14 pages, 2 figures. A preliminary version appeared in SoCG 2012.
This version removes an incorrect example at the end of Section 3.
Quantitative Tverberg, Helly, & Carath\'eodory theorems
This paper presents sixteen quantitative versions of the classic Tverberg,
Helly, & Caratheodory theorems in combinatorial convexity. Our results include
measurable or enumerable information in the hypothesis and the conclusion.
Typical measurements include the volume, the diameter, or the number of points
in a lattice.Comment: 33 page
Algorithmic and Hardness Results for the Colorful Components Problems
In this paper we investigate the colorful components framework, motivated by
applications emerging from comparative genomics. The general goal is to remove
a collection of edges from an undirected vertex-colored graph such that in
the resulting graph all the connected components are colorful (i.e., any
two vertices of the same color belong to different connected components). We
want to optimize an objective function, the selection of this function
being specific to each problem in the framework.
We analyze three objective functions, and thus, three different problems,
which are believed to be relevant for the biological applications: minimizing
the number of singleton vertices, maximizing the number of edges in the
transitive closure, and minimizing the number of connected components.
Our main result is a polynomial time algorithm for the first problem. This
result disproves the conjecture of Zheng et al. that the problem is -hard
(assuming ). Then, we show that the second problem is -hard,
thus proving and strengthening the conjecture of Zheng et al. that the problem
is -hard. Finally, we show that the third problem does not admit
polynomial time approximation within a factor of for
any , assuming (or within a factor of , assuming ).Comment: 18 pages, 3 figure
Balanced Families of Perfect Hash Functions and Their Applications
The construction of perfect hash functions is a well-studied topic. In this
paper, this concept is generalized with the following definition. We say that a
family of functions from to is a -balanced -family
of perfect hash functions if for every , , the number
of functions that are 1-1 on is between and for some
constant . The standard definition of a family of perfect hash functions
requires that there will be at least one function that is 1-1 on , for each
of size . In the new notion of balanced families, we require the number
of 1-1 functions to be almost the same (taking to be close to 1) for
every such . Our main result is that for any constant , a
-balanced -family of perfect hash functions of size can be constructed in time .
Using the technique of color-coding we can apply our explicit constructions to
devise approximation algorithms for various counting problems in graphs. In
particular, we exhibit a deterministic polynomial time algorithm for
approximating both the number of simple paths of length and the number of
simple cycles of size for any
in a graph with vertices. The approximation is up to any fixed desirable
relative error
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