4. Matchings (a) For this question, use the bipartite graph in Figure 4. a b W X d y e Ꮓ f Figure 4: Bipartite graph i. Clearly indicate on the graph the matching M = {(b,v), (c,z), (e, w),(f,y)}. ii. Overlay on the appropriate network clearly indicating the source, the sink and the edge capacities. iii. Use Ford-Fulkerson to find a larger matching M'. iv. Run Ford-Fulkerson one more time, then as in lectures, use the run to find a vertex cover of size 5 (Question 4 continued on next page) (Question 4 continued) (b) Suppose the companies Anthropic, Google, Meta and Microsoft are looking to hire the development teams of the Large Language Models (LLM's) ChatGPT, Claude, Gemini and Llama. The companies have the following preference list (from highest to lowest) Anthropic Google Meta Microsoft Claude Gemini ChatGPT Llama Claude Gemini ChatGPT Llama Claude ChatGPT Llama ChatGPT Gemini Llama Gemini Claude and the teams (of developers of the large language models) have the following preference list (again highest to lowest) ChatGPT Microsoft Anthropic Meta Google Claude Anthropic Google Meta Gemini Google Anthropic Meta Llama Meta Microsoft Google Microsoft Microsoft Anthropic i. With the companies proposing to the teams of developers, find a stable matching.

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