Consider the following task of building a zoo for Drexel! You’ve gathered m donors to fund the creation of your new park, and you’ve picked the location, so now you just need to choose the inhabitants. Ideally you’d get every animal imaginable, except you just don’t have space. As it is, you have room to comfortably fit k animals. Since you’d like to ensure that your funding doesn’t dry up, you figure your donors should get to make requests about which animals will be kept. After sending out a few emails, you collect from each donor i a list of animals Ai which donor i would like to have at the zoo. As you suspected, the total number of different animals appearing on the m lists exceeds k, so you won’t be able to satisfy all of their requests. You decide the fairest thing to do is to ensure that at least 1 animal from each Ai is chosen. Given all these lists, we want to know whether we can select a set H of at most k animals to put in the zoo such that each donor i will be able to see at least one of the animals from her list Ai (that is, H ∩ Ai is non-empty for each i). Prove that this problem is NP-complete.
Consider the following task of building a zoo for Drexel! You’ve gathered m donors to fund the creation of
your new park, and you’ve picked the location, so now you just need to choose the inhabitants. Ideally you’d
get every animal imaginable, except you just don’t have space. As it is, you have room to comfortably fit
k animals. Since you’d like to ensure that your funding doesn’t dry up, you figure your donors should get
to make requests about which animals will be kept. After sending out a few emails, you collect from each
donor i a list of animals
of different animals appearing on the m lists exceeds k, so you won’t be able to satisfy all of their requests.
You decide the fairest thing to do is to ensure that at least 1 animal from each Ai
is chosen.
Given all these lists, we want to know whether we can select a set H of at most k animals to put in the
zoo such that each donor i will be able to see at least one of the animals from her list Ai (that is, H ∩ Ai
is
non-empty for each i).
Prove that this problem is NP-complete.
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