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Found in: Page 343

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# A stable assignment, defined in the preamble to Exercise 60 in Section 3.1, is called optimal for suitors if no stable assignment exists in which a suitor is paired with a suitee whom this suitor prefers to the person to whom this suitor is paired in this stable assignment. Use strong induction to show that the deferred acceptance algorithm produces a stable assignment that is optimal for suitors.

The deferred acceptance algorithm produces a stable assignment that is optimal for suitors. The statement holds true for $n=1$ number of suitors and it also holds true for number of suitors. The statement also holds true $n=k+1$ for suitors of $n=1,2,...,k$.

See the step by step solution

## Step 1: Significance of the strong induction

The strong induction is mainly described as a proof which is related to the simple induction. The strong induction is mainly used for proving a particular theorem.

## Step 2: Determination of the deferred acceptance algorithm

Let $P\left(n\right)$ is described as a statement which states that “deferred acceptance algorithm produces a stable assignment that is optimal for suitors, when there are number of suitees and number of suitors”.

In the basic step where $n=1$, as there are only suitee and suitors, the suitor has been assigned by the stable assignment to the particular suitee and the particular assignment is also optimal for a particular suitor. Hence, $P\left(1\right)$ holds true.

In the inductive step, let the series $P\left(1\right),P\left(2\right),....,P\left(k\right)$ holds true, then it is to be considered that $P\left(k+1\right)$ is also true. Let, the number of suitees and suitors are $k+1$ respectively. If the $k+1$ suitors are being divided into two different groups such as Q and p, then the algorithm of deferred acceptance mainly produces a stable assignment which is mainly optimal for the and also suitors which states that $P\left(1\right),P\left(2\right),....,P\left(k\right)$ holds true. Hence, $P\left(k+1\right)$ holds true.

Thus, the deferred acceptance algorithm produces a stable assignment that is optimal for suitors. The statement holds true for $n=1$ number of suitors and it also holds true for $n=k+1$ number of suitors. The statement also holds true for suitors of $n=1,2,...,k$.