On Metric Multi-Covering Problems

2016

We consider the metric multi-cover problem (MMC). The input consists of 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 $\alpha \geq 1$. 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 $\alpha$-th powers of the radii of the balls. Bar-Yehuda[2013] gave a $3^{\alpha} \cdot k$-approximation for the MMC problem. Bhowmick et al [SoCG 13, JoCG 15] showed that an $O(1)$ approximation (independent of the coverage demand $k$) exists for the special case in which $X, Y$ are points in a fixed-dimensional space $\mathbb{R}^d$. In this paper, we present an $O(1)$-approximation for the MMC problem in any arbitrary metric space, improving the result of Bar-Yehuda[2013] and matching the guar...

Algorithmica, 2020

In this article, we study some fault-tolerant covering problems in metric spaces. In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space (X ∪ Y , d), a positive integer k that represents the coverage demand of each client, and a constant α ≥ 1. Each server can host a single ball of arbitrary radius centered on it. Each client x ∈ 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. We also study some non-trivial generalizations of the MMC, 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. We present the first constant approximations for these fault-tolerant covering problems. Our algorithms are based on the following paradigm: for each of the three 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. The reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain our results for the MMC and these generalizations.

2013

We consider 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 → N, and a constant α ≥ 1. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client x ∈ 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.

Lecture Notes in Computer Science, 2011

We present a packing-based approximation algorithm for the k-Set Cover problem. We introduce a new local search-based k-set packing heuristic, and call it Restricted k-Set Packing. We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted k-Set Packing algorithm, our k-Set Cover algorithm is composed of the k-Set Packing heuristic [7] for k ≥ 7, Restricted k-Set Packing for k = 6, 5, 4 and the semi-local (2, 1)improvement [2] for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of H k − 0.6402 + Θ(1 k), where H k is the k-th harmonic number. For small k, our results are 1.8667 for k = 6, 1.7333 for k = 5 and 1.5208 for k = 4. Our algorithm improves the currently best approximation ratio for the k-Set Cover problem of any k ≥ 4.

2018

Lecture Notes in Computer Science, 2015

We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications (e.g., reduce the communication cost among sensors) in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation algorithm, assuming that the ratio between the sensor radius and communication radius is bounded. In Budgeted-CSC problem, our goal is to choose a set of B sensors, such that the number of targets covered by the chosen sensors is maximized and the communication graph is connected. We also obtain a constant approximation under the same assumption.

arXiv (Cornell University), 2009

2011

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph.

Operations Research Letters, 2009

A dual type algorithm constructs the minimum covering ball of a given finite set of points in R n by finding the minimum covering balls of a sequence of subsets, each with no more than n + 1 points and with strictly increasing radius, until all points are covered. (P.M. Dearing). center, without violating the covering property, until optimality is reached. Elzinga and Hearn [4] developed a dual approach in which the minimum covering circle is found for a sequence of subsets S ⊆ P, each of at most 3 points and with increasing radius, until some circle covers the entire set P. Other approaches use Voronoi diagrams . Meggido [6] developed a theoretical linear time algorithm for solving the problem.

Proceedings. 13th International Conference on Computer Communications and Networks (IEEE Cat. No.04EX969), 2000

In overdeployed sensor networks, one approach to conserve energy is to keep only a small subset of sensors active at any instant. In this article, we consider the problem of selecting a minimum size connected-cover, which is defined as a set of sensors ¡ such that each point in the sensor network is "covered" by at least different sensors in ¡ , and the communication graph induced by ¡ is connected. For the above optimization problem, we design a centralized approximation algorithm that delivers a near-optimal (within a factor of ¢ ¤ £ ¦ ¥ § ©) solution, and present a distributed version of the algorithm. We also present a communication-efficient localized distributed algorithm which is empirically shown to perform well.