A novel convex dual approach to three-dimensional assignment problem: theoretical analysis

In this paper, we propose a novel convex dual approach to the three dimensional assignment problem, which is an NP-hard binary programming problem. It is shown that the proposed dual approach is equivalent to the Lagrangian relaxation method in terms of the best value attainable by the two approaches. However, the pure dual representation is not only more elegant, but also makes the theoretical analysis of the algorithm more tractable. In fact, we obtain a sufficient and necessary condition for the duality gap to be zero, or equivalently, for the Lagrangian relaxation approach to find the optimal solution to the assignment problem with a guarantee. Also, we establish a mild and easy-to-check condition, under which the dual problem is equivalent to the original one. In general cases, the optimal value of the dual problem can provide a satisfactory lower bound on the optimal value of the original assignment problem. Furthermore, the newly proposed approach can be extended to higher dimensional cases and general assignment problems.