Minimum-Cost Coverage of Point Sets by Disks

We consider a class of geometric facility location problems in which the goal is to determine a set X of disks given by their centers (t_j) and radii (r_j) that cover a given set of demand points Y in the plane at the smallest possible cost. We consider cost functions of the form sum_j f(r_j), where f(r)=r^alpha is the cost of transmission to radius r. Special cases arise for alpha=1 (sum of radii) and alpha=2 (total area); power consumption models in wireless network design often use an exponent alpha>2. Different scenarios arise according to possible restrictions on the transmission centers t_j, which may be constrained to belong to a given discrete set or to lie on a line, etc. We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points t_j on a given line in order to cover demand points Y in the plane; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover Y; (c) a proof of NP-hardness for a discrete set of transmission points in the plane and any fixed alpha>1; and (d) a polynomial-time approximation scheme for the problem of computing a minimum cost covering tour (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.Comment: 10 pages, 4 figures, Latex, to appear in ACM Symposium on Computational Geometry 200

Approximation Algorithms for Geometric Covering Problems for Disks and Squares

Geometric covering is a well-studied topic in computational geometry. We study three covering problems: Disjoint Unit-Disk Cover, Depth-(≤ K) Packing and Red-Blue Unit-Square Cover. In the Disjoint Unit-Disk Cover problem, we are given a point set and want to cover the maximum number of points using disjoint unit disks. We prove that the problem is NP-complete and give a polynomial-time approximation scheme (PTAS) for it. In Depth-(≤ K) Packing for Arbitrary-Size Disks/Squares, we are given a set of arbitrary-size disks/squares, and want to find a subset with depth at most K and maximizing the total area. We prove a depth reduction theorem and present a PTAS. In Red-Blue Unit-Square Cover, we are given a red point set, a blue point set and a set of unit squares, and want to find a subset of unit squares to cover all the blue points and the minimum number of red points. We prove that the problem is NP-hard, and give a PTAS for it. A "mod-one" trick we introduce can be applied to several other covering problems on unit squares

Target Assignment in Robotic Networks: Distance Optimality Guarantees and Hierarchical Strategies

We study the problem of multi-robot target assignment to minimize the total distance traveled by the robots until they all reach an equal number of static targets. In the first half of the paper, we present a necessary and sufficient condition under which true distance optimality can be achieved for robots with limited communication and target-sensing ranges. Moreover, we provide an explicit, non-asymptotic formula for computing the number of robots needed to achieve distance optimality in terms of the robots' communication and target-sensing ranges with arbitrary guaranteed probabilities. The same bounds are also shown to be asymptotically tight. In the second half of the paper, we present suboptimal strategies for use when the number of robots cannot be chosen freely. Assuming first that all targets are known to all robots, we employ a hierarchical communication model in which robots communicate only with other robots in the same partitioned region. This hierarchical communication model leads to constant approximations of true distance-optimal solutions under mild assumptions. We then revisit the limited communication and sensing models. By combining simple rendezvous-based strategies with a hierarchical communication model, we obtain decentralized hierarchical strategies that achieve constant approximation ratios with respect to true distance optimality. Results of simulation show that the approximation ratio is as low as 1.4