On the completability of orthogonal Latin rectangles

Abstract : We address the problem of completability for 2-row orthogonal Latin rectangles (OLR2). The approach is to identify all incomplete pairs of 2-row Latin rectangles that are not completable to an OLR2 and are minimal with respect to this property; i.e., we characterize all circuits of the independence system associated with OLR2. Since there can be no polytime algorithm generating the clutter of circuits of an arbitrary independence system, our work adds to the few such cases for which that clutter is fully described. The result has a direct polyhedral implication; it gives rise to inequalities that are valid for the polytope associated with orthogonal Latin squares and thus planar multi-dimensional assignment. A complexity result is also at hand: completing an incomplete set of (n-1) MOLR2 is NP-complete.