Minimum-cost coverage of point sets by disks

Abstract

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 ⊂ R 2 at the smallest possible cost. We consider cost functions of the form ∑ j f (r j ), where f (r) = r α is the cost of transmission to radius r. Special cases arise for α = 1 (sum of radii) and α = 2 (total area); power consumption models in wireless network design often use an exponent α > 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 ⊂ R 2 ; (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 R 2 and any fixed α > 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.

Sensor planning for a symbiotic UAV and UGV system for precision agriculture

“…The robots have to separately query each sensor, unlike our case, where the soil measurements for multiple points can be combined by sampling in the intersection of their neighborhoods. Alt et al [12] studied the problem of covering a given set of points with k radio antenna with circular ranges by choosing the center and radius r i for each circle. The cost function is a weighted sum of the length of the tour and the sum of r α i for each disk (α models the radio transmission power).…”

Energy-Efficient User Association in Broadcast Transmission

“…Subsequently, we introduce a new algorithm based on Markovian approximation, called distributed BS selection algorithm. We refer to [2][3][4][5][6][7] for further reading as related literature to our work.…”

Maximum Area Independent Sets in Disk Intersection Graphs

“…We illustrate with one example: Given a set of potential wireless network servers at specified locations and with associated disk covering ranges, find a subset of servers that maximize the covered area under the constraint of non-interference. See also the paper by Alt et al 2 for recent results on covering point sets by disks. MIS in unit disk graphs is known to be NP-hard 9 , which immediately implies the NP-hardness of MWIS and MAIS in general disk intersection graphs.…”