Question

(The 'harem problem') Let \(B\) be a set of boys, and suppose that each boy in \(B\) wishes to marry more than one of his girl friends. Find a necessary and sufficient condition for the harem problem to have a solution. (Hint: replace each boy by several identical copies of himself, and then use Hall's theorem.)

Short answer

The necessary and sufficient condition for the harem problem to have a solution is: For any subset of boys, the number of girls that those boys like is at least equal to the number of boys.

Step-by-step solution

Understanding the problem

The problem talks about a set \(B\) of boys and each boy in \(B\) wants to marry more than one of his girlfriends. The task is to find a necessary and sufficient condition for this harem problem to have a solution.

Application of Hall's theorem

Given the hint, each boy in the set \(B\) is replaced by several identical copies of himself, one for each girl he likes. This situation forms a bipartite graph. Now, Hall's theorem can be applied. Hall's theorem states that for any set X of vertices in one part of the graph, the number of vertices in the neighbourhood of X is at least equal to the number of vertices in X.

Condition for the harem problem

In this problem, the above statement of Hall's theory translates to: For any subset of boys, the number of girls that those boys like is at least equal to the number of boys. This is the necessary and sufficient condition for the problem to have a solution, i.e., for each boy to marry more than one girl.

Key concepts

Graph Theory

At its core, Graph Theory is a field of mathematics and computer science focused on the properties of graphs, which are structures used to model pairwise relations between objects. A graph is composed of vertices (or nodes) and edges (or lines) that connect them.

Why is this important? Graphs come into play in numerous real-world scenarios, from social networks where individuals are vertices and their relationships are edges, to transportation networks with locations as vertices and routes as edges. In the context of the exercise, each boy and each girl can be considered as vertices, and their potential marital relationship as edges, forming what is known as a bipartite graph.

Bipartite Graph

A Bipartite Graph is a special kind of graph where vertices can be divided into two disjoint sets, such that every edge connects a vertex from one set to a vertex in the other set; there are no edges between vertices of the same set. For example, in a job fair, one set could represent companies, and the other set could represent job seekers, with edges symbolizing potential employment.

In the harem problem, the two sets would be boys and girls, with edges indicating a boy's wish to marry a girl. Understanding how to model the problem using a bipartite graph is crucial because it sets the stage for applying Hall's Theorem, a fundamental principle that determines the possibility of a perfect matching in such graphs.

Harem Problem

The Harem Problem, as illustrated in the exercise, is an example of how graph theory can be employed to solve complex relationship-based problems. It is a playful name for a serious mathematical model that explores matching within bipartite graphs.

To solve it, we imagine creating copies of each boy for every girl he wishes to marry. We then apply Hall's Theorem which, in essence, requires that for any subset of boys, the collective number of girls they are interested in must be at least as large as the number of boys in that subset. If this condition is met across all subsets, it's theoretically possible to have a satisfying arrangement where every boy can marry more than one girl. This principle is not just theoretical but can be applied to various allocation and assignment problems in areas such as economics, resource distribution, and network design.