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Kasturi Varadarajan

Publications and source records attributed to Kasturi Varadarajan.

17 recordsLinked to original sources
arXiv2021

Non-Uniform $k$-Center and Greedy Clustering

In the Non-Uniform $k$-Center 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 $t$-NU$k$C Problem, 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 $1 \le i \le t$. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for $t$-NU$k$C is not possible if $t$ is unbounded. 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 $t$-NU$k$C 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 $3$-NU$k$C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [To appear in 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 $t$-NU$k$C problem, eventually bringing us closer to the resolution of the CGK conjecture.

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arXiv2019

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 $ρ$, such that with $k$ balls of radius $ρ$, 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.

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arXiv2018

On the Partition Set Cover Problem

Several algorithms with an approximation guarantee of $O(\log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into $r$ \emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $\mathcal{C}_t$, using the minimum number of sets. We give a randomized LP-rounding algorithm that is an $O(β+ \log r)$ approximation for the Partition Set Cover problem. Here $β$ denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which $β$ is known to be sublogarithmic in $n$. We also extend the LP rounding algorithm to obtain $O(\log r)$ approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an $o(\log r)$-approximation for any of these problems.

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arXiv2017

Capacitated Covering Problems in Geometric Spaces

In this article, we consider the following capacitated covering problem. We are given a set $P$ of $n$ points and a set $\mathcal{B}$ of balls from some metric space, and a positive integer $U$ that represents the capacity of each of the balls in $\mathcal{B}$. We would like to compute a subset $\mathcal{B}' \subseteq \mathcal{B}$ of balls and assign each point in $P$ to some ball in $\mathcal{B}$ that contains it, such that the number of points assigned to any ball is at most $U$. The objective function that we would like to minimize is the cardinality of $\mathcal{B}$. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions $3$ and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for these problems. In fact, in the Euclidean setting, only $(1+ε)$ factor expansion is sufficient for any $ε> 0$, with the approximation factor being a polynomial in $1/ε$. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.

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arXiv2017

On Partial Covering For Geometric Set Systems

We study a generalization of the Set Cover problem called the \emph{Partial Set Cover} in the context of geometric set systems. The input to this problem is a set system $(X, \mathcal{S})$, where $X$ is a set of elements and $\mathcal{S}$ is a collection of subsets of $X$, and an integer $k \le |X|$. The goal is to cover at least $k$ elements of $X$ by using a minimum-weight collection of sets from $\mathcal{S}$. The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for a certain projection of the set system $(X, \mathcal{S})$. As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems.

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arXiv2017

On Metric Multi-Covering Problems

In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X \cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $α\geq 1$. Each server can have a single ball of arbitrary radius centered on it. Each client $x \in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $α$-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the $t$-MMC, where we require the number of open servers to be at most some given integer $t$. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for $1$-covering, we obtain the first constant approximations for the MMC and these generalizations.

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arXiv2016

Approximate Clustering via Metric Partitioning

In this paper we consider two metric covering/clustering problems - \textit{Minimum Cost Covering Problem} (MCC) and $k$-clustering. In the MCC problem, we are given two point sets $X$ (clients) and $Y$ (servers), and a metric on $X \cup Y$. We would like to cover the clients by balls centered at the servers. The objective function to minimize is the sum of the $α$-th power of the radii of the balls. Here $α\geq 1$ is a parameter of the problem (but not of a problem instance). MCC is closely related to the $k$-clustering problem. The main difference between $k$-clustering and MCC is that in $k$-clustering one needs to select $k$ balls to cover the clients. For any $\eps > 0$, we describe quasi-polynomial time $(1 + \eps)$ approximation algorithms for both of the problems. However, in case of $k$-clustering the algorithm uses $(1 + \eps)k$ balls. Prior to our work, a $3^α$ and a ${c}^α$ approximation were achieved by polynomial-time algorithms for MCC and $k$-clustering, respectively, where $c > 1$ is an absolute constant. These two problems are thus interesting examples of metric covering/clustering problems that admit $(1 + \eps)$-approximation (using $(1+\eps)k$ balls in case of $k$-clustering), if one is willing to settle for quasi-polynomial time. In contrast, for the variant of MCC where $α$ is part of the input, we show under standard assumptions that no polynomial time algorithm can achieve an approximation factor better than $O(\log |X|)$ for $α\geq \log |X|$.

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arXiv2015

On Variants of k-means Clustering

\textit{Clustering problems} often arise in the fields like data mining, machine learning etc. to group a collection of objects into similar groups with respect to a similarity (or dissimilarity) measure. Among the clustering problems, specifically \textit{$k$-means} clustering has got much attention from the researchers. Despite the fact that $k$-means is a very well studied problem its status in the plane is still an open problem. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound in polynomial time is $9+\eps$. In this paper, we consider the following variant of $k$-means. Given a set $C$ of points in $\mathcal{R}^d$ and a real $f > 0$, find a finite set $F$ of points in $\mathcal{R}^d$ that minimizes the quantity $f*|F|+\sum_{p\in C} \min_{q \in F} {||p-q||}^2$. For any fixed dimension $d$, we design a local search PTAS for this problem. We also give a "bi-criterion" local search algorithm for $k$-means which uses $(1+\eps)k$ centers and yields a solution whose cost is at most $(1+\eps)$ times the cost of an optimal $k$-means solution. The algorithm runs in polynomial time for any fixed dimension. The contribution of this paper is two fold. On the one hand, we are being able to handle the square of distances in an elegant manner, which yields near optimal approximation bound. This leads us towards a better understanding of the $k$-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that very little is known about the local search method for geometric approximation.

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arXiv2015

A Constant Factor Approximation for Orthogonal Order Preserving Layout Adjustment

Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e.\ maintains the sorted ordering of the rectangle centers along both $x$-axis and $y$-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles(LADR). It was known that LADR is $\mathbb{NP}$-hard, but only heuristics were known for it. We show that a certain decision version of LADR is $\mathbb{APX}$-hard, and give a constant factor approximation for LADR.

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arXiv2014

A Constant-Factor Approximation for Multi-Covering with Disks

We consider variants of the following multi-covering problem with disks. We are given two point sets $Y$ (servers) and $X$ (clients) in the plane, a coverage function $κ:X \rightarrow \mathcal{N}$, and a constant $α\geq 1$. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client $x \in X$ be covered by at least $κ(x)$ of the server disks. The objective function we wish to minimize is the sum of the $α$-th powers of the disk radii. We present a polynomial time algorithm for this problem achieving an $O(1)$ approximation.

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arXiv2012

On the Sensitivity of Shape Fitting Problems

In this article, we study shape fitting problems, $ε$-coresets, and total sensitivity. We focus on the $(j,k)$-projective clustering problems, including $k$-median/$k$-means, $k$-line clustering, $j$-subspace approximation, and the integer $(j,k)$-projective clustering problem. We derive upper bounds of total sensitivities for these problems, and obtain $ε$-coresets using these upper bounds. Using a dimension-reduction type argument, we are able to greatly simplify earlier results on total sensitivity for the $k$-median/$k$-means clustering problems, and obtain positively-weighted $ε$-coresets for several variants of the $(j,k)$-projective clustering problem. We also extend an earlier result on $ε$-coresets for the integer $(j,k)$-projective clustering problem in fixed dimension to the case of high dimension.

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arXiv2011

On Isolating Points Using Disks

In this paper, we consider the problem of choosing disks (that we can think of as corresponding to wireless sensors) so that given a set of input points in the plane, there exists no path between any pair of these points that is not intercepted by some disk. We try to achieve this separation using a minimum number of a given set of unit disks. We show that a constant factor approximation to this problem can be found in polynomial time using a greedy algorithm. To the best of our knowledge we are the first to study this optimization problem.

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arXiv2010

Algorithms and Hardness for Subspace Approximation

The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm instead. Most of the previous work on subspace approximation has focused on small or constant $k$ and $p$, using coresets and sampling techniques from computational geometry. In this paper, extending another line of work based on convex relaxation and rounding, we give a polynomial time algorithm, \emph{for any $k$ and any $p \geq 2$}, with the approximation guarantee roughly $γ_{p} \sqrt{2 - \frac{1}{n-k}}$, where $γ_{p}$ is the $p$-th moment of a standard normal random variable N(0,1). We show that the convex relaxation we use has an integrality gap (or "rank gap") of $γ_{p} (1 - ε)$, for any constant $ε> 0$. Finally, we show that assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than $γ_{p} (1 - ε)$, for any constant $ε> 0$.

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arXiv2009

Quasi-Polynomial Time Approximation Schemes for Target Tracking

We consider the problem of tracking $n$ targets in the plane using $2n$ cameras. We can use two cameras to estimate the location of a target. We are then interested in forming $n$ camera pairs where each camera belongs to exactly one pair, followed by forming a matching between the targets and camera pairs so as to best estimate the locations of each of the targets. We consider a special case of this problem where each of the cameras are placed along a horizontal line $l$, and we consider two objective functions which have been shown to give good estimates of the locations of the targets when the distances between the targets and the cameras are sufficiently large. In the first objective, the value of an assignment of a camera pair to a target is the tracking angle formed by the assignment. Here, we are interested in maximizing the sum of these tracking angles. A polynomial time 2-approximation is known for this problem. We give a quasi-polynomial time algorithm that returns a solution whose value is at least a $(1-ε)$ factor of the value of an optimal solution for any $ε> 0$. In the second objective, the cost of an assignment of a camera pair to a target is the ratio of the vertical distance between the target and $l$ to the horizontal distance between the cameras in the camera pair. In this setting, we are interested in minimizing the sum of these ratios. A polynomial time 2-approximation is known for this problem. We give a quasi-polynomial time algorithm that returns a solution whose value is at most a $(1+ε)$ factor of the value of an optimal solution for any $ε> 0$.

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arXiv2009

Decomposing Coverings and the Planar Sensor Cover Problem

We show that a $k$-fold covering using translates of an arbitrary convex polygon can be decomposed into $Ω(k)$ covers (using an efficient algorithm). We generalize this result to obtain a constant factor approximation to the sensor cover problem where the ranges of the sensors are translates of a given convex polygon. The crucial ingredient in this generalization is a constant factor approximation algorithm for a one-dimensional version of the sensor cover problem, called the Restricted Strip Cover (RSC) problem, where sensors are intervals of possibly different lengths. Our algorithm for RSC improves on the previous $O(\log \log \log n)$ approximation.

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arXiv2005

Improved Approximation Algorithms for Geometric Set Cover

Given a collection S of subsets of some set U, and M a subset of U, the set cover problem is to find the smallest subcollection C of S such that M is a subset of the union of the sets in C. While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually U = R^d. Extending prior results, we show that approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R of S and function f(), there is a decomposition of the portion of U not covered by R into an expected f(|R|) regions, each region of a particular simple form. We show that under this condition, a cover of size O(f(|C|)) can be found. Our proof involves the generalization of shallow cuttings to more general geometric situations. We obtain constant-factor approximation algorithms for covering by unit cubes in R^3, for guarding a one-dimensional terrain, and for covering by similar-sized fat triangles in R^2. We also obtain improved approximation guarantees for fat triangles, of arbitrary size, and for a class of fat objects.

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