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EECS 340/454: Algorithms
Assignment 8 (454 Only): NP-Completeness Problem 1 The Maximum-Disjoint-Subset problem is defined as follows: We are given a collection S of sets. We would like to find the maximum number of disjoint sets in S. (a) Formulate the decision version of the Maximum-Disjoint-Subset problem. (b) Prove that the decision version you formulated in (a) is NP-complete by reduction from 3-CNF-SAT. Problem 2 Consider the following decision problems: • 0-1 Knapsack: Given a finite set S of n items, weight function wi : S → Z, value function vi : S → Z, and two integers W and V , is there a subset C ⊆ S such that ∑ i∈C wi ≤W and∑ i∈C vi ≥ V ? • Subset-Sum: Given a set R = {b1, b2, ...., bm} of m integers and another integer B, is there a subset D ⊆ R such that ∑i∈D bi = B? Given that Subset-Sum is NP-complete, prove that 0-1 Knapsack is NP-Complete.