Approximating the multi-level bottleneck assignment problem

We consider the multi-level bottleneck assignment problem (MBA). This problem is described in the recent book "Assignment Problems" by Burkard et al. (2009) on pages 188-189, where its complexity status is called open. We view the problem as a special case of a bottleneck m-dimensional multi-index assignment problem and settle its complexity status. We give approximation algorithms and inapproximability results, depending upon the completeness of the underlying graph