This paper presents fast approximation algorithms for the knapsack and sum of subset problems. The authors propose an algorithm that finds an approximate solution to the 0/1 knapsack problem within a relative error ε (0 < ε < 1) in O(n log n) time and O(n) space. The algorithm is modified to handle the unbounded knapsack problem, where the δ_i's can be any nonnegative integer, resulting in O(n) computing time. Additionally, a linear-time algorithm is obtained for a special class of 0/1 knapsack problems where the profit density p_i/c_i is the same for all items. The algorithm is also adapted for the unbounded knapsack problem, achieving O(n) time complexity. The paper also discusses the approximation algorithms for other forms of the knapsack problem, including the sum of subset problem, and shows that there exist P-complete problems with efficient ε-approximate algorithms for any ε < 1. The results contrast with previous findings that some P-complete problems are also P-complete for approximate solutions. The paper concludes that while the algorithms are efficient, they require significant space, making them impractical for small ε values. The authors acknowledge the contributions of the referees for improving the paper's presentation.This paper presents fast approximation algorithms for the knapsack and sum of subset problems. The authors propose an algorithm that finds an approximate solution to the 0/1 knapsack problem within a relative error ε (0 < ε < 1) in O(n log n) time and O(n) space. The algorithm is modified to handle the unbounded knapsack problem, where the δ_i's can be any nonnegative integer, resulting in O(n) computing time. Additionally, a linear-time algorithm is obtained for a special class of 0/1 knapsack problems where the profit density p_i/c_i is the same for all items. The algorithm is also adapted for the unbounded knapsack problem, achieving O(n) time complexity. The paper also discusses the approximation algorithms for other forms of the knapsack problem, including the sum of subset problem, and shows that there exist P-complete problems with efficient ε-approximate algorithms for any ε < 1. The results contrast with previous findings that some P-complete problems are also P-complete for approximate solutions. The paper concludes that while the algorithms are efficient, they require significant space, making them impractical for small ε values. The authors acknowledge the contributions of the referees for improving the paper's presentation.