You have established your vegan hot-dog chain, but now you need to get some vegan hot dog chefs in the kitchens. You have n chefs to train, and each must take two classes: one on bun toasting and one on sauce squirting. Kostas is the best bun-toaster in San Francisco. He has agreed to teach your chefs how to toast buns. He has scheduled several different sessions which your chefs may attend, though the maximum attendance of each session may vary. Let B be the set of bun-toasting classes offered and let bk be the number of chefs that can attend the k-th class. Additionally, you have paid Kostas to train exactly n students, so the sum of bk over all of Kostas' classes is exactly n. Wei-Lin is the best sauce-squirter in San Francisco. His deal with you is the same as Kostas's. We will denote Mikes's classes as S and let sk represent the number of students that can attend his k-th class. Again, the sum of sk over all of Mike's classes is exactly n. Each student gives you two lists Bi ⊆ B and Si ⊆ S of which classes they are available to attend. Give a polynomial time algorithm that assigns each student to a bun-toasting and a sauce-squirting class for which they are available, or proves that no such assignment is possible.
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