Brief Announcement: Pan and Scan
ABSTRACT We introduce the pan and scan problem, in which cameras are configured to observe multiple target locations. A camera's configuration consists of its orientation and its zoom factor or field or view (its position is given); the quality of a target's reading by a camera depends (inversely) on both the distance and field of view. After briefly discussing an easy setting in which a target accumulates measurement quality from all cameras observing it, we move on to a more challenging setting in which for each target only the best measurement of it is counted, for which we give various results. Although both variants admit continuous solutions, we observe that we may restrict our attention to solutions based on pinned cones. For a geometrically constrained setting, we give an optimal dynamic programming algorithm. For the unconstrained setting of this problem, we prove NPhardness, present efficient centralized and distributed 2approximation algorithms, and observe that a PTAS exists under certain assumptions. For a synchronized distributed setting, we give a 2approximation protocol and a (2β)/(1α)approximation protocol (for all 0 ≤ α ≤ 1 and β ≥ 1) with the stability feature that no target's camera assignment changes more than logβ(m/α) times. We also discuss the running times of the algorithms and study the speedups that are possible in certain situations.
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Conference Paper: Maximum Coverage Problem with Group Budget Constraints and Applications
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ABSTRACT: We study a variant of the maximum coverage problem which we label the maximum coverage problem with group budget constraints (MCG). We are given a collection of sets S = fS1;S2;:::;Smg where each set Si is a subset of a given ground set X. In the maximum cov erage problem the goal is to pick k sets fromS to maximize the cardi nality of their union. In the MCG problemS is partitioned into groups G1;G2;:::;G'. The goal is to pick k sets fromS to maximize the car dinality of their union but with the additional restriction that at most one set be picked from each group. We motivate the study of MCG by pointing out a variety of applications. We show that the greedy algorithm gives a 2approximation algorithm for this problem which is tight in the oracle model. We also obtain a constant factor approximation algorithm for the cost version of the problem. We then use MCG to obtain the rst constant factor approximation algorithms for the following problems: (i) multiple depot ktraveling repairmen problem with covering constraints and (ii) orienteering problem with time windows when the number of time windows is a constant.Approximation, Randomization, and Combinatorial Optimization, Algorithms and Techniques, 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2004, and 8th International Workshop on Randomization and Computation, RANDOM 2004, Cambridge, MA, USA, August 2224, 2004, Proceedings; 01/2004  [Show abstract] [Hide abstract]
ABSTRACT: We prove that (1 + o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1+o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. 1 Introduction Let S be a set of n points and F = fS 1 ; S 2 ; : : : Sm g a collection of subsets of S. Set cover is the problem of selecting as few as possible subsets from F such that every point in S is contained in at least one of the selected subsets. This problem is NPhard, but can be approximated within a ratio of ln n, where ln denotes the natural logarithm. Lund and Yannakakis [14] showed that it is hard to approximate set cover within a ratio of (log n)=2, where log denotes logarithms in base 2. We extend their hardness result, and show that for any ffl...Journal of the ACM 05/1999; 45(4). DOI:10.1145/285055.285059 · 1.39 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Sensor nodes may be equipped with a "directional" sensing device (such as a camera) which senses a physical phenomenon in a certain direction depending on the chosen orientation. In this article, we address the problem of selection and orientation of such directional sensors with the objective of maximizing coverage area. Prior works on sensor coverage have largely focused on coverage with sensors that are associated with a unique sensing region. In contrast, directional sensors have multiple sensing regions associated with them, and the orientation of the sensor determines the actual sensing region. Thus, the coverage problems in the context of directional sensors entails selection as well as orientation of sensors needed to activate in order to maximize/ensure coverage. In this article, we address the problem of selecting a minimum number of sensors and assigning orientations such that the given area (or set of target points) is kcovered (i.e., each point is covered k times). The above problem is NPcomplete, and even NPhard to approximate. Thus, we design a simple greedy algorithm that delivers a solution that kcovers at least half of the target points using at most M log(kC) sensors, where C is the maximum number of target points covered by a sensor and M is the minimum number of sensor required to kcover all the given points. The above result holds for almost arbitrary sensing regions. We design a distributed implementation of the above algorithm, and study its performance through simulations. In addition to the above problem, we also look at other related coverage problems in the context of directional sensors, and design similar approximation algorithms for them.Sensor, Mesh and Ad Hoc Communications and Networks, 2009. SECON '09. 6th Annual IEEE Communications Society Conference on; 07/2009