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The knapsack problem (KP) can be solved with dynamic programming (DP) in O(nS), where n is the number of items and 5 the knapsack capacity. O Such a complexity is not polynomial with respect to the size the instance when a binary encoding is used. Thus, the algorithm is not (strictly) polynomial O if the instance is encoded in unary, such a complexity is polynomial. Thus, one can claim that the DP algorithm is polynomial in the general case. O Since the problem is NP-hard, it cannot be solved in polynomial time even if the instance is encoded in unary, unless P-NP.

The knapsack problem (KP) can be solved with dynamic programming (DP) in O(nS), where n is the number of items and 5 the knapsack capacity. O Such a complexity is not polynomial with respect to the size the instance when a binary encoding is used. Thus, the algorithm is not (strictly) polynomial O if the instance is encoded in unary, such a complexity is polynomial. Thus, one can claim that the DP algorithm is polynomial in the general case. O Since the problem is NP-hard, it cannot be solved in polynomial time even if the instance is encoded in unary, unless P-NP.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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Question 1 The knapsack problem (KP) can be solved with dynamic programming (DP) in O(nS), where n is the number of items and S the knapsack capacity. O Such a complexity is not polynomial with respect to the size the instance when a binary encoding is used. Thus, the algorithm is not (strictly) polynomial O f the instance is encoded in unary, such a complexity is polynomial. Thus, one can claim that the DP algorithm is polynomial in the general case. O Since the problem is NP-hard, it cannot be solved in polynomial time even if the instance is encoded in unary. unless P-NP.
Transcribed Image Text:Question 1 The knapsack problem (KP) can be solved with dynamic programming (DP) in O(nS), where n is the number of items and S the knapsack capacity. O Such a complexity is not polynomial with respect to the size the instance when a binary encoding is used. Thus, the algorithm is not (strictly) polynomial O f the instance is encoded in unary, such a complexity is polynomial. Thus, one can claim that the DP algorithm is polynomial in the general case. O Since the problem is NP-hard, it cannot be solved in polynomial time even if the instance is encoded in unary. unless P-NP.
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