On the completability of incomplete orthogonal Latin rectangles

We address the problem of completability for 2-row orthogonal Latin rectangles (OLR2). Our approach is to identify all pairs of incomplete 2-row Latin rectangles that are not com- pletable 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 poly- time algorithm generating the clutter of circuits of an arbitrary independence system, our work adds to the few independence systems 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 assign- ment. A complexity result is also at hand: completing a set of (n - 1) incomplete MOLR2 is NP-complete

On the completability of mutually orthogonal Latin rectangles

This thesis examines the completability of an incomplete set of m-row orthogonal Latin rectangles (MOLRm) from a set theoretical viewpoint. We focus on the case of two rows, i.e. MOLR2, and define its independence system (IS) and the associated clutter of bases, which is the collection of all MOLR2. Any such clutter gives rise to a unique clutter of circuits which is the collection of all minimal dependent sets. To decide whether an incomplete set of MOLR2 is completable, it suffices to show that it does not contain a circuit therefore full knowledge of the clutter of circuits is needed. For the IS associated with 2-row orthogonal Latin rectangles (OLR2) we establish a methodology based on the notion of an availability matrix to fully characterise the corresponding clutter of circuits. We prove that..

Testing of random matrices

Let $n$ be a positive integer and $X = [x_{ij}]_{1 \leq i, j \leq n}$ be an $n \times n$\linebreak \noindent sized matrix of independent random variables having joint uniform distribution \hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k \leq n} = \frac{1}{n} \quad (1 \leq i, j \leq n) \koz. A realization $\mathcal{M} = [m_{ij}]$ of $X$ is called \textit{good}, if its each row and each column contains a permutation of the numbers $1, 2,..., n$. We present and analyse four typical algorithms which decide whether a given realization is good

Completing partial Latin squares with two filled rows and two filled columns

It is shown that any partial Latin square of order at least six which consists of two filled rows and two filled columns can be completed