Identifying the Pareto-Optimal Solutions for Multi-point Distance Minimization Problem in Manhattan Space

Multi-point distance minimization problems (M-DMP) pose a number of theoretical challenges and simultaneously represent a number of practical applications, particularly in navigational and layout design problems. When the Euclidean distance measure is minimized for each target point, the resulting problem has a trivial solution, however such a consideration limits its application in practice. Since two-dimensional landscapes, floor spaces, or printed circuit boards are laid out in gridded structures for convenience, M-DMP problems are to be solved with Manhattan distance metric for their practical significance. Identification of Pareto-optimal solutions leading to trade-off minimal paths from multiple target points become a challenging task. In this paper, we suggest a systematic procedure for identifying Pareto-optimal solutions and provide a theoretical proof for validating our construction process. Thus, instead of devising generic multi-objective optimization algorithms for solving M-DMP problem under Manhattan space, which has been demonstrated to be a challenging task, this paper advocates the use of a computationally fast construction procedure. Further practicalities of the M-DMP problem are outlined and it is argued that future applications must involve a basic theoretical construction step similar to the procedure suggested here along with efficient algorithmic methods for handling those practicalities.