One key element in the construction of the fully polynomial-time approximation scheme for the knapsack problem was the ability to compute lower and upper bounds for the optimal value that are within a factor of n of each other (using the maximum value piece that fits in the knapsack to get the lower bound). Use the result of the previous exercise to derive a refined approximation scheme that eliminates one factor of n in the running time of the algorithm.

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What best describes a career pathway in a lodging career? a worker starts out as an amusement attendant, then becomes a recreation worker, and then becomes a gaming worker within five years. a worker starts out as a bell hop, then becomes a night clerk, and then becomes a hotel manager within five years. a worker starting out as a tour guide, then becomes a travel clerk, and then becomes a travel agent within five years. a worker starts out as a server, then becomes a food preparer, and then becomes a head chef within five years.

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