Results 1  10
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27
MAXIMUM AREA INDEPENDENT SETS IN DISK INTERSECTION GRAPHS
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2008
"... Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are wellknown problems in combinatorial optimization; they are NPhard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restric ..."
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Cited by 6 (6 self)
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Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are wellknown problems in combinatorial optimization; they are NPhard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constantratio approximation algorithms for finding an independent set with a large total area relative to the union area.
Cheap or Flexible Sensor Coverage
"... Abstract. We consider dual classes of geometric coverage problems, in which disks, corresponding to coverage regions of sensors, are used to cover a region or set of points in the plane. The first class of problems involve assigning radii to alreadypositioned sensors (being cheap). The second class ..."
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Cited by 5 (4 self)
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Abstract. We consider dual classes of geometric coverage problems, in which disks, corresponding to coverage regions of sensors, are used to cover a region or set of points in the plane. The first class of problems involve assigning radii to alreadypositioned sensors (being cheap). The second class of problems are motivated by the fact that the sensors may, because of practical difficulties, be positioned with only approximate accuracy (being flexible). This changes the character of some coverage problems that solve for optimal disk positions or disk sizes, ordinarily assuming the disks can be placed precisely in their chosen positions, and motivates new problems. Given a set of disk sensor locations, we show for most settings how to assign either (near)optimal radius values or allowable amounts of placement error. Our primary results are 1) in the 1d setting we give a faster dynamic programming algorithm for the (linear) sensor radius problem; and 2) we find a maxmin fair set of radii for the 2d continuous problems in polynomial time. We also give results for other settings, including fast approximation algorithms for the 1d continuous case. 1
Sensor Planning for a Symbiotic UAV and UGV system for Precision Agriculture
"... Abstract — We study the problem of coordinating an Unmanned Aerial Vehicle (UAV) and an Unmanned Ground Vehicle (UGV) for a precision agriculture application. In this application, the ground and aerial measurements are used for estimating nitrogen (N) levels ondemand across a farm. Our goal is to ..."
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Abstract — We study the problem of coordinating an Unmanned Aerial Vehicle (UAV) and an Unmanned Ground Vehicle (UGV) for a precision agriculture application. In this application, the ground and aerial measurements are used for estimating nitrogen (N) levels ondemand across a farm. Our goal is to estimate the N map over a field and classify each point based on N deficiency levels. These estimates in turn guide fertilizer application. Applying the right amount of fertilizer at the right time can drastically reduce fertilizer usage. Towards building such a system, this paper makes the following contributions: First, we present a method to identify points whose probability of being misclassified is above a threshold. Second, we study the problem of maximizing the number of such points visited by an UAV subject to its energy budget. The novelty of our formulation is the capability of the UGV to mule the UAV to deployment points. This allows the system to conserve the short battery life of a typical UAV. Third, we introduce a new path planning problem in which the UGV must take a measurement within a disk centered at each point visited by the UAV. The goal is to minimize the total time spent in traveling and measuring. For both problems, we present constantfactor approximation algorithms. Finally, we demonstrate the utility of our system with simulations which use manually collected soil measurements from the field. I.
Connecting a set of circles with minimum sum of radii
 In Proceedings of the 12th international conference on Algorithms and data structures, WADS’11
, 2011
"... Abstract. We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, ..."
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Cited by 4 (0 self)
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Abstract. We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NPhard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomialtime algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomialtime approximation schemes, and conclude with experimental results and open problems.
Power assignment problems in wireless communication: Covering points by disks, reaching few receivers quickly, and energyefficient travelling salesman tours
 In Proc. 4th Int. IEEE Conf. Distributed Comput. Sensor Systems, LNCS 5067
, 2008
"... Abstract. A fundamental class of problems in wireless communication is concerned with the assignment of suitable transmission powers to wireless devices/stations such that the resulting communication graph satisfies certain desired properties and the overall energy consumed is minimized. Many concre ..."
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Abstract. A fundamental class of problems in wireless communication is concerned with the assignment of suitable transmission powers to wireless devices/stations such that the resulting communication graph satisfies certain desired properties and the overall energy consumed is minimized. Many concrete communication tasks in a wireless network like broadcast, multicast, pointtopoint routing, creation of a communication backbone, etc. can be regarded as such a power assignment problem. This paper considers several problems of that kind; the first problem was studied before in [1,6] and aims to select and assign powers to k out of a total of n wireless network stations such that all stations are within reach of at least one of the selected stations. We show that the problem can be (1+) approximated by only looking at a small subset of the input, which is of size O k
A note on multicovering with disks
 Comput. Geom
"... In theDisk Multicover problem the input consists of a set P of n points in the plane, where each point p ∈ P has a covering requirement k(p), and a set B of m base stations, where each base station b ∈ B has a weight w(b). If a base station b ∈ B is assigned a radius r(b), it covers all points in th ..."
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Cited by 3 (0 self)
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In theDisk Multicover problem the input consists of a set P of n points in the plane, where each point p ∈ P has a covering requirement k(p), and a set B of m base stations, where each base station b ∈ B has a weight w(b). If a base station b ∈ B is assigned a radius r(b), it covers all points in the disk of radius r(b) centered at b. The weight of a radii assignment r: B → R is defined as b∈B w(b)r(b) α, for some constant α. A feasible solution is an assignment such that each point p is covered by at least k(p) disks, and the goal is to find a minimum weight feasible solution. The Polygon Disk Multicover problem is a closely related problem, in which the set P is a polygon (possibly with holes), and the goal is to find a minimum weight radius assignment that covers each point in P at least K times. We present a 3αkmaxapproximation algorithm for Disk Multicover, where kmax is the maximum covering requirement of a point, and a (3αK + ε)approximation algorithm for Polygon Disk Multicover.
Multiobjective Disk Cover Admits a PTAS
"... We introduce multiobjective disk cover problems and study their approximability. We construct a polynomialtime approximation scheme (PTAS) for the multiobjective problem where k types of points (customers) in the plane have to be covered by disks (base stations) such that the number of disks is min ..."
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Cited by 1 (0 self)
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We introduce multiobjective disk cover problems and study their approximability. We construct a polynomialtime approximation scheme (PTAS) for the multiobjective problem where k types of points (customers) in the plane have to be covered by disks (base stations) such that the number of disks is minimized and for each type of points, the number of covered points is maximized. Our approximation scheme can be extended so that it works with the following additional features: interferences, different services for different types of customers, different shapes of supply areas, weighted customers, individual costs for base stations, and payoff for the quality of the obtained service. Furthermore, we show that it is crucial to solve this problem in a multiobjective way, where all objectives are optimized at the same time. The constrained approach (i.e., the restriction of a multiobjective problem to a single objective) often used for such problems can significantly degrade their approximability. We can show nonapproximability results for several singleobjective restrictions of multiobjective disk cover problems. For example, if there are 2 types of customers, then maximizing the supplied customers of one type is not even approximable within a constant factor, unless P = NP. 1
A faster dynamic programming algorithm for facility location
 In FWCG 2006
"... We study the following 1D and 1.5D problems: Given a set of m server locations {t1,...,tm}, lying along the xaxis and specified in increasing order, and n client locations {p1,..., pn}, also specified in increasing order along the ..."
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We study the following 1D and 1.5D problems: Given a set of m server locations {t1,...,tm}, lying along the xaxis and specified in increasing order, and n client locations {p1,..., pn}, also specified in increasing order along the
Distributed Gateway Placement for Cost Minimization in Wireless Mesh Networks
 INTERNATIONAL CONFERENCE ON DISTRIBUTED COMPUTING SYSTEMS
, 2010
"... We study the problem of gateway placement for cost minimization (GPCM) in twodimensional wireless mesh networks. We are given a set of mesh routers, assume they have identical transmission range r, represented by unit transmission disks around them. A router may be selected as a gateway at certain ..."
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We study the problem of gateway placement for cost minimization (GPCM) in twodimensional wireless mesh networks. We are given a set of mesh routers, assume they have identical transmission range r, represented by unit transmission disks around them. A router may be selected as a gateway at certain placing cost. A router is served by a gateway if and only if the gateway is within its transmission range. The goal of this work is to select a set of mesh routers as gateways to serve the rest routers with minimum overall cost. This problem is NPhard. To the best of our knowledge, no distributed algorithm with a constant approximation ratio has been given before. When all weights are uniform, the best approximation ratio is 38. We present both centralized and distributed algorithms which can achieve approximation ratios 6+ɛ and 20 respectively. Our algorithms greatly improve the best approximation ratios.
A constantfactor approximation for multicovering with disks
 In Proceedings of the Symposium on Computational Geometry (SoCG
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