AMS/MBA 546 (Fall, 2010)
Estie Arkin
Network Flows: Solution sketch to homework set # 9
1). Modify the input by adding

M

W

“±ctional women”, so that now the number of men is equal to the
number of women. Each man adds the ±ctional women to the bottom of his preference list. The preference
lists for the ±ctional women is arbitrary. Now apply the algorithm of men proposing described in class (AMO
12.5), the result is a stable marriage for the original men and the enlarged set of women, by the theorem
from class. Now remove the ±ctional women, and some men become single. We claim that this is a stable
marriage for the original problem. First note that all women will remain married. Now, consider a man
m
and a (non ±ctional!) woman
w
not married to each other in
M
. If man
m
is not married, the algorithm
had him married earlier to a ±ctional woman. This means he proposed and got rejected by all real women,
including
m
, who rejected him in favor of someone she prefers. If
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 Fall '08
 Arkin,E
 Graph Theory, Planar graph, Estie Arkin, planar subgraph, maximal planar subgraph

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