Assignment problems are among the best studied combinatorial optimization
problems. Typically, such a problem arises whenever the members of a first set have to be allocated/mapped to those of a second one in such a way that the overall cost is minimized. Studying k-tuples instead of pairs over k sets of equal size leads to multi-index assignment problems, in particular (k,s)-assignment problems, the parameter s indicating that every s-tuple of elements (each from a different set) is assigned to a (k-s)-tuple. For (k,s)=(2,1), for instance, we get back the classical 2-index assignment problem, and for s=1 or s=2 we obtain the family of axial or planar assignment problems, respectively. In the planar case and if k=3, any solution corresponds to a latin square, and if k=4 we come to mutually orthogonal latin squares (MOLS).
Assignment problems cover a large spectrum of applications:
to cite just a few. Our work on assignment problems goes back to the 1980's, when we studied planar 3-dimensional assignment problems with respect to their solvability by linear programming techniques, in other words: from a polyhedral point of view. At the same time we initiated a study on the completability of incomplete latin squares with the objective to describe those structures that could give an answer to the question: when is an incomplete latin square completable. Relations with class-teacher time table problems have also been investigated.
Our current work is on the completability of a particular type of incomplete latin rectangles and on open questions related to the existence of MOLS.
More complete information can be obtained here.
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R.Euler, On the completability of incomplete latin squares,
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R.Euler, When is an incomplete (3,n)-Latin rectangle completable?
Rapport technique, Lab-STICC UMR CNRS 3192,
mars 2011, invited presentation: 24th European Conference on Operational
Research, July 2010, Lisbon, and: 5th World Conference on 21st Century
Mathematics, February 2011, Lahore, submitted.