On the solvability of the localization problem in robot networks

This paper contributes to the problem of deterministic localization of robot networks using local and relative observations only. This is an important issue in collective and cooperative robotics where global positioning systems are not available, and the basic premise is the localization ability of the group. We prove that, giving a set of relative observations made by the robots, the unique non ambiguous pose estimation of the robot network in a deterministic way, is a NP-hard problem. This means that no polynomial-time algorithm can deterministically solve the unique pose estimation problem based on relative observations. The consequence is that no guaranty can be provided, in a polynomial time, that the possibly estimated poses of the robots, will correspond to the effective (actual) ones. The proof is based on complexity theory. We build appropriate polynomial-time reductions acting on the localization problem and leading to well known NP-hard problems. The paper gives some tracks to overcome this issue.