Polyhedral Methods for Solving Three Index Assignment Problems

The (axial) three index assignment problem, also known as the three-dimensional matching problem, is the problem of assigning one item to one job at one point or interval of time in such a way as to minimize the total cost of the assignment. Until now the most efficient algorithms explored for solving this problem are based on polyhedral combinatorics. So far, four important facet classes Q, P, B and C have been characterized and O(n3 )(linear-time) separation algorithms for five facet subclasses of Q, P and B have been established. The complexity of these separation algorithms is best possible since the number of the variables of three index assignment problem of order n is n3. In this paper, we review these progresses and raise some further questions on this topic.