On facets of the three-index assignment polytope

Geena Gwan and Liqun Qi School of Mathematics The University of New South Wales Kensington, N.S.W. 2033 Australia Two new classes of facet-defining inequalities for the three-index assignrnent polytope are identified in this paper. According to the shapes of their support index sets, we call these facets bull facets and comb facets respectively. The bull facet has Chvatal rank 1, while the comb facet has Chvatal rank 2. For a comb facet-defining inequality, the right-hand-side coefficient is a positive integer, and the left-ha,nd-side coefficients equal to 0 or 1. For a bull facet-defining inequality, the right-hand-side coefficient is a positive even integer, and the left-hand-side coefficients equal to 0, 1 or 2. Furthermore, we give an O(n 3 ) procedure for finding a bull facet with the right-hand-side coefficient 2, violated by a given noninteger solution to the linear programming relaxation of the three-index assignment problem, or showing that no such facet exists. Such an algorithm is called a separation algorithm. Since the number of variables is n 3 and one needs to check through all the variables in such a separation algorithm, this algorithm is linear-time and the order of its complexity is the best possible.