Implementation of a 3-D assignment algorithm in MATLAB

The unique assignment problem for three data sets (or 3-D assignment problem) is NP-hard, meaning that an optimal solution cannot be solved in polynomial time. This research explores some of the issues associated with implementing and testing a Lagrangian relaxation-based approximate assignment algorithm. The issues associated with solving a non-square (3-D) cost matrix are discussed along with the formulation of the multitarget tracking problem as a 3-D assignment problem. An optimal algorithm via exhaustive search (cost matrix smallest dimension <3) was written to test the performance of the Lagrangian relaxation algorithm. Dimensionality was limited because computation time for the exhaustive search algorithm grows with (m·n·p)³, where m, n, and p are the dimensions of the 3-D cost matrix. For small dimensions, the Lagrangian relaxation method was found to often give the optimal solution; it always gave a reasonably "good" solution.