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# 数学代写|线性规划代写Linear Programming代考|MAT3100 The Integrality Theorem

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## 数学代写|线性规划代写Linear Programming代考|The Integrality Theorem

In this section, we consider network flow problems for which all the supplies and demands are integers. Such problems are called network flow problems with integer data. As we explained in Section 14.2, for network flow problems, basic primal solutions are computed without any multiplication or division. The following important theorem follows immediately from this property:

THEOREM 14.2. Integrality Theorem. For network flow problems with integer data, every basic feasible solution and, in particular, every basic optimal solution assigns integer flow to every arc.

This theorem is important because many real-world network flow problems have integral supplies/demands and require their solutions to be integral too. This integrality restriction typically occurs when one is shipping indivisible units through a network. For example, it wouldn’t make sense to ship one third of a car from an automobile assembly plant to one dealership with the other two thirds going to another dealership.

Problems that are linear programming problems with the additional stipulation that the optimal solution values must be integers are called integer programming problems. Generally speaking, these problems are much harder to solve than linear programming problems (see Chapter 23). However, if the problem is a network flow problem with integer data, it can be solved efficiently using the simplex method to compute a basic optimal solution, which the integrality theorem tells us will be integer valued.

## 数学代写|线性规划代写Linear Programming代考|Konig’s Theorem

In addition to its importance in real-world optimization problems, the integrality theorem also has many applications to the branch of mathematics called combinatorics. We illustrate with just one example.

THEOREM 14.3. König’s Theorem. Suppose that there are $n$ girls and $n$ boys, that every girl knows exactly $k$ boys, and that every boy knows exactly $k$ girls. Then $n$ marriages can be arranged with everybody knowing his or her spouse.

Before proving this theorem it is important to clarify its statement by saying that the property of “knowing” is symmetric (for example, knowing in the biblical sense). That is, if a certain girl knows a certain boy, then this boy also knows this girl.

PROoF. Consider a network with nodes $g_1, g_2, \ldots, g_n, b_1, b_2, \ldots, b_n$ and an arc from $g_i$ to $b_j$ if girl $i$ and boy $j$ know each other. Assign one unit of supply to each girl node and a unit of demand to each boy node. Assign arbitrary objective coefficients to create a well-defined network flow problem. The problem is guaranteed to be feasible: just put a flow of $1 / k$ on each arc (the polygamists in the group might prefer this nonintegral solution). By the integrality theorem, the problem has an integer-valued solution. Clearly, the flow on each arc must be either zero or one. Also, each girl node is the tail of exactly one arc having a flow of one. This arc points to her intended mate.

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