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On the Gotlieb-Csima time-tabling algorithm

M. A. H. Dempster

1968
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Canadian Journal of Mathematics - Journal Canadien de Mathematiques
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1. Introduction. This paper concerns an algorithm, proposed by C. C. Gotlieb (4) and modified by J. Csima (1; 2), for a recent combinatorial problem whose application includes the construction of school time-tables. Theoretically, the problem is related to systems of distinct subset representatives, the construction of Latin arrays, the colouring of graphs, and flows in networks (1; 2; 3). It was conjectured by Gotlieb and Csima that if solutions to a given time-table problem existed, i.e. if
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... existed, i.e. if time-tables incorporating certain pre-assigned meetings existed, their algorithm would find one. In the contrary case, it would indicate which pre-assignments were incompatible with the remainder. A computer-generated counterexample to this conjecture has recently been reported by J. Lions (6), and it is the purpose of this paper to analyse the situation in detail. It is known that in the absence of pre-assignments, a time-table can always be found (1). Suppose a non-negative integral n X n matrix R = (p tj ) is given whose row and column sums are all m. Then one may form an initial 0-1 availability array A 0 = (a°i jk ) for the time-table problem with requirements matrix R, T(R), as follows. Consider as a 3-dimensional array, a stack of m n X n 0-1 matrices A°k whose non-zero entries correspond to those of R. In school time-tabling, p t > y = 4, for example, expresses a requirement for teacher i' to meet class j' for four periods in the w-period day. The unit entries a°i'y k , k = 1, . . . , m, of A 0 represent possible meetings and express the obvious fact that before scheduling is begun, teacher i' and class j' are available to meet in any four of the m periods of the day. Define the union of two 0-1 arrays X = (% ijk ) and Y = (rj ijk ) of comparable dimensions to be the 0-1 array and their intersection to be the 0-1 array Define these concepts similarly for 0-1 matrices and vectors of comparable dimensions. For example, if X = (1010101) and Y = (0111011), then XU Y = (1111111) andin7= (0010001). A 0-1 array (or matrix or vector) X may be said to be contained in a 0-1 array (or matrix or vector) Y of comparable dimensions, denoted X C Y, if X P\ Y = X. An entry % ijk of

doi:10.4153/cjm-1968-013-7
fatcat:6cnarteldfc33ksqcowiqr7zbe