Tree-based algorithms for the multidimensional assignment problem

The Multidimensional Assignment Problem (MAP) is an extension of the two-dimensional assignment problem in which we wish to find an optimal matching of elements between mutually exclusive sets. Although the two-dimensional assignment problem has been shown to be solvable in polynomial time, extending the dimensions to three makes the problem NP-complete. The multidimensional assignment problem has many practical applications including the data association problem. This work investigates the application of Greedy Randomized Adaptive Search Procedures (GRASP) and branch and bound algorithms based on two different tree representations of the MAP. The first representation of the MAP is an index-based tree which is derived from the 0-1 integer programming formulation. Every level of the tree represents a different value of the first index. The second representation comes from the permutation formulation of the MAP and is referred to as a permutation-based tree. Each level in this tree represents a different permutation vector. The number of dimensions and the number of elements in each dimension will affect the effectiveness of the algorithms. We investigate the advantages and disadvantages of using either tree to perform GRASP and branch and bound algorithm.