Question

4314. In a large university with k academic departments, we must appoint an impor- tant committee. One professor will be chosen from each department. Some professors have joint appointments in two or more departments, but each must be the designated representative of at most one department. We must use equally many assistant profes- sors, associate professors, and full professors among the chosen representatives (assume that& is divisible by 3). How can the committee be found? (Hint: Build a network in which units of flow correspond to professors chosen for the committee and capacities en- force the various constraints. Explain how to use the network to test whether such a committee exists and find it if it does.) (Hall [19561)

Answer 1

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aga in a unlt capoci ty nodes node is Sent to 0f PYDfessor Yonk Thus, jomn every Yonk, on edge capacf ty Kla is Present Sink t to Evesy Pofessox can be in one deportment only as the makes thot edgos Source SuTe Onigina ting tsdm he choosen once aRow than eveyy depaytment connot be bu the unrt choosen is Committee A Pofessox foy the yanks the vepresenting of the in the the low . hus and balanced is equal an edges Present feasible vałve of the sink. oni when the into mannes Committe is formed and this t. the flow is k. the is these in the Capacity of k. when a minlmum Possible cot having the SauYCe - Sink. cuts have the capacity infinity edge when the Cuts the will vesult S Pooved Thus, the vequived