Optimal Perimeter Guarding with Heterogeneous Robot Teams: Complexity Analysis and Effective Algorithms

We perform structural and algorithmic studies of significantly generalized versions of the optimal perimeter guarding (OPG) problem. As compared with the original OPG where robots are uniform, in this paper, many mobile robots with heterogeneous sensing capabilities are to be deployed to optimally guard a set of one-dimensional segments. Two complimentary formulations are investigated where one limits the number of available robots (OPG{}_{LR}) and the other seeks to minimize the total deployment cost ({\rm OPG}_{MC}). In contrast to the original OPG which admits low-polynomial time solutions, both OPG{}_{LR} and {\rm OPG}_{MC} are computationally intractable with OPG{}_{LR} being strongly NP-hard. Nevertheless, we develop fairly scalable pseudo-polynomial time algorithms for practical, fixed-parameter subcase of OPG{}_{LR}; we also develop pseudo-polynomial time algorithm for general {\rm OPG}_{MC} and polynomial time algorithm for the fixed-parameter {\rm OPG}_{MC} case. The applicability and effectiveness of selected algorithms are demonstrated through extensive numerical experiments.