sol9 - AMS/MBA 546(Fall 2010 Estie Arkin Network Flows...

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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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