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Unformatted text preview: CS601: HW 4 Solutions 3 Sept, 2009 1 (a) Prove Independent Set ≤ p Diverse Subset Sol. Given an instance of Independent Set problem, a graph G = ( V, E ), we can convert to an instance of Diverse Subset problem as follows: • For each vertex v i add a customer c i . • For each edge e i add a product p i . • If v i and v j contain an edge e k , then denote customers c i and c j as having bought the item p k . There are exactly V × E entries in the matrix, hence takes polynomial time to construct. 1 (b) Prove Diverse Subset ≤ p Independent Set Sol. Given the Customer-Product matrix, the corresponding instance of independent set can be built as follows: • For each customer c i add a vertex v i . • Connect v i and v j with an edge if they happened to buy a common product. This can be done in linear time( O ( P )). The overall procedure take C 2 × P time. 1 (c) Prove that Diverse Subset ≤ p Circuit-SAT Sol. As discussed in class, we merely have to build a circuit that checks if in a given set of customersSol....
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- Summer '09