Optimization problems in communication networks

We study four problems arising in the area of communication networks. The minimum-weight dominating set problem in unit disk graphs asks, for a given set D of weighted unit disks, to find a minimum-weight subset D' ⊆ D such that the disks D' intersect all disks D. The problem is NP-hard and we present the first constant-factor approximation algorithm. Applying our techniques to other geometric graph problems, we can obtain better (or new) approximation algorithms. The network discovery problem asks for a minimum number of queries that discover all edges and non-edges of an unknown network (graph). A query at node v discovers a certain portion of the network. We study two different query models and show various results concerning the complexity, approximability and lower bounds on competitive ratios of online algorithms. The OVSF-code assignment problem deals with assigning communication codes (nodes) from a complete binary tree to users. Users ask for codes of a certain depth and the codes have to be assigned such that (i) no assigned code is an ancestor of another assigned code and (ii) the number of (previously) assigned codes that have to be reassigned (in order to satisfy (i)) is minimized. We present hardness results and several algorithms (optimal, approximation, online and fixed-parameter tractable). The joint base station scheduling problem asks for an assignment of users to base stations (points in the plane) and for an optimal colouring of the resulting conflict graph: user u with its assigned base station b is in conflict with user v, if a disk with center at 6, and u on its perimeter, contains v. We study the complexity, and present and analyse optimal, approximation and greedy algorithms for general and various special cases.