Decentralized algorithms for optimally rigid network constructions

In this paper, we address the construction of optimally rigid networks that minimize an edge-weighted objective function over a planar graph. We propose two auction-based algorithms to solve this problem in a fully decentralized way. The first approach finds an optimal solution at the cost of high communication complexity; the second approach provides a sub-optimal solution while reducing the computational burden according to a sliding mode parameter ζ, yielding a tradeoff between complexity and optimality. A theoretical characterization of the optimality of the first algorithm is provided, and a closed form for the maximum gap between the optimal solution and the sub-optimal solution is also given. Simulation results are presented to corroborate the theoretical findings.