On the role of symmetry in solving maximum lifetime problem in two-dimensional sensor networks

We analyze a continuous and discrete symmetries of the maximum lifetime problem in two dimensional sensor networks. We show, how a symmetry of the network and invariance of the problem under a given transformation group $G$ can be utilized to simplify its solution. We prove, that for a $G$-invariant maximum lifetime problem there exists a $G$-invariant solution. Constrains which follow from the $G$-invariance allow to reduce the problem and its solution to a subset, an optimal fundamental region of the sensor network. We analyze in detail solutions of the maximum lifetime problem invariant under a group of isometry transformations of a two dimensional Euclidean plane.