Given a set of targets that need to be monitored and a vehicle, we consider a combinatorial motion planning problem where the objective is to find a path for the vehicle such that each target is visited at least once by the vehicle, the path satisfies the motion constraints of the vehicle and the length of the path is a minimum. This is an NP-hard problem and currently, there are no algorithms that can find an optimal solution to this problem. In this article, we model the motion of the vehicle as a Dubins car and develop a method that can provide tight lower bounds to the motion planning problem. We accomplish this by relaxing the constraints corresponding to the angle of approach at each of the targets and then penalizing them whenever they are violated. The solution to the Lagrangian relaxation gives a lower bound, and this lower bound is maximized over the penalty variables using subgradient optimization. The proposed method is the first of its kind for finding tight lower bounds for combinatorial motion planning problems and can be extended to similar problems with more general motion constraints.

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