The Marriage Lemma for Polygamists

Abstract

It has long been the ambition of one of the authors (it is left as an exercise to the reader to determine which one) to appear as a guest on one of the sensational TV talk shows. But what could a mathematician talk about that would be of interest to Geraldo or Donahue? The first subject that comes to mind is the so-called marriage problem, alias the assignment problem; here the objective is to have a matching between a set of women and a set of men given that each woman likes some of the men and dislikes the rest. Let the integer m represent the number of women; the number of men must, of course, be at least m. An obvious necessary condition that each woman gets herself a man is that for each p < m, every set of p women must together like at least p different men. Philip Hall's theorem, which has come to be known as the Marriage Lemma, asserts that this obvious necessary condition is sufficient. The proof of this result can most easily be accomplished by expressing the problem as that of determining the maximum flow in an appropriate network and then applying the maximum flow minimum cut theorem [9]. Another method of proof is based on the concept of maximum matchings in bipartite graphs that was developed by D. Konig and Philip Hall [1]. After further reflection, the authors suddenly had their moment of inspiration: Why not generalize the result to the polygamous case, i.e., where each man can be married to more than one woman? To be specific, for a fixed positive integer d ? 1 we assume that up to d women can be married to one man. In the monogamous case we clearly needed at least m men, but now the only obvious condition is that the number n of men must satisfy m < nd. Note that we do not require each man to marry d women; a complete matching requires only that all women are married. Note also that we do not require the compatibility to be symmetric. A woman may be married to a man if she likes him, even if he does not like her.