Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs

Abstract

For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimumweight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.Our techniques also yield a constant-factor approximation algorithm for the weighted disk cover problem (covering a set of points in the plane with unit disks of minimum total weight) and a 3-approximation algorithm for the weighted forwarding set problem (covering a set of points in the plane with weighted unit disks whose centers are all contained in a given unit disk).

Maximising lifetime for fault-tolerant target coverage in sensor networks

“…This relationship also shows that the problem is N P-hard [6]. A series of papers has presented approximation algorithms with smaller and smaller constant approximation ratios for weighted set cover with unit disks [1,13,7,9,20]. The currently best known ratio is 4 + ε [9,20].…”

“…These results apply also to the minimum-weight dominating set problem in unit disk graphs. If one is interested in a minimum-weight connected dominating set, a standard approach is to first compute a cheap dominating set and then solve a node-weighted Steiner tree problem to connect it [1,7,9,13,20]. The node-weighted Steiner tree problem admits a 2.5α-approximation algorithm in unit disk graphs [10,19], where α is the approximation ratio of the best known approximation algorithm for edge-weighted Steiner trees.…”

“…See Figure 2 for an illustration. 1 Note that this sandglass is uniquely determined by p l and pr and there are O(|Pij| 2 ) possibilities for guessing p l and pr. We show that the coverage requirements of any point of P ij located in the 2-watching t l sandglass for lm are met by disks from upper ∪ lower.…”

Wireless coverage with disparate ranges

“…Note that, a lot of recent algorithmic results [2,6,7,14] for min-weight dominating set imply algorithms for min-cost disk cover with the same approximation ratios.…”

Energy and Time Efficient Broadcasting in Known Topology Radio Networks

“…In this model one can provide almost immediately an approximate solution based on the efficient computation of minimal connected dominating sets with logarithmic approximation ratio in general graphs [15] and constant approximation ratio in unit disk graphs, see e.g. [2]. An interesting related open problem is to look for trade-offs between the total number of transmissions and the broadcasting time.…”