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comp360_hw4sol

# comp360_hw4sol - Problem 1 We define our supermarket...

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Problem 1 We define our supermarket problem as an NP problem: Problem : SUPER Input : A list of customers, and the items they have purchased, and an integer l. Question : Is there a subset S of customers of size l such that no two customers in S have bought the same item? First, we note that this problem is indeed in NP. If there is such a subset of customers, then the subset itself is our certificate. We simply go through the items each customer has purchased, and if check that there are no duplicates. This takes O (number of items). We will show that SUPER is NP-complete. First, we select a problem known to be NP-complete, the Independent Set problem: Problem : IND SET Input : A graph G = ( V , E ), integer k . Question : Is there a subset S of k vertices so that no two vertices in S share an edge? To show SUPER is NP-complete, we reduce any instance of IND SET to an instance of SUPER, in polynomial time. Reduction : Given a graph G and integer k as input to IND SET, we create an instance of SUPER as follows: For each vertex in G , create a customer in SUPER. For each edge in G create an item in SUPER. A customer has bought an item if and only if its corresponding vertex is incident to the item's corresponding edge. Let l = k . e.g. Customers Items bought 1 1,2,3 2 1 3 4 4 2 5 3,4,5 6 5 Clearly this can be done in polynomial time. We can do this by a Breadth First Search.

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comp360_hw4sol - Problem 1 We define our supermarket...

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